Could Quark model turn out to be false? Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. 
Due to a phenomenon known as color confinement, quarks are never directly observed or found in isolation; they can be found only within baryons or mesons.
This sentence makes me very nervous: Due to a phenomenon known as color confinement
This sentence is Like I want to prove something spurious to save the subject (quark).
 A: It is believed for good reason that confinement isn't just some made up thing.  The good reason is from Lattice QCD.  While it is not analytic proof, simulations of QCD show that QCD is indeed confining naturally.  That is, confinement is already hidden inside QCD and happens naturally. Furthermore, for high enough temperatures QCD is deconfining, and the result is a quark-gluon plasma, where the two run around like a plasma. It is unfortunate that as of yet little experimental evidence exists for a deconfining phase of QCD, at least to my knowledge.  But computational work on QCD sheds a great deal of insight into the genuine physics of QCD.  As a note, deep inelastic scattering provides evidence for quarks, as the experiment resembles Rutherford scattering, which phenomenologically helped develop an accurate model of the atom eventually.  Funny enough the "planet model" of the atom is an inaccurate interpretations in certain circumstances.
That being said, our interpretations of physics and physical systems changes (period).  So almost inevitably, while quarks will probably stick around, how we view them will change.
If you are interested in constraints put on QCD by computational simulations look into Lattice QCD and Gauge Fields on a Lattice.  It has been an important step in tapping into non-perturbative QCD.
A: Your position on quarks, that they are made-up constituents, was the mainstream view in physics from 1964, when Zweig and Gell-Mann independently proposed quarks, to November 1974, when the charm quark was discovered and everyone else decided it was the right idea. In this intermediate time, the idea of a permanently confined particle was considered suspect, so the theories of the strong interaction were not allowed to speak about hypothetical pointlike consituents. The quark model was still very strongly supported by indirect evidence, but in a sense it is good that strong interaction community rejected these ideas, otherwise we might never have string theory.
The phenomenon of confinement is not so mysterious, it was already understood on 1 dimension by Schwinger in the early 1960s. In 1 dimension of space, the electric force doesn't spread out as it does in 3 dimensions, and the force doesn't get weaker with distance. This means if you pull apart an electron and an anti-electron in 1d, at some point, you do enough work to make new electron anti-electron pair from the vacuum, and this pair-production neutralizes the two particles. This means that the only finite energy states are neutral composites. This was extended to the non-abelian case by 'tHooft, and the notion of confinement is completely understood in 1+1 dimensions.
The idea Gell-Mann indirectly promoted, and which was established by many people in the 1970s, is that something like this happens with quarks, that they are linked by a flux-string that makes the force constant, as if it were effectively 1 dimensional. This flux-tube idea was not well established in the 1970s, but you can't reject it anymore. Aside from lattice simulations, which show the flux tube in static-force calculations (the force between two quarks is constant with distance when they are far apart, just as in 1 dimension), there are also known exact dualities between string theories of infinitesimal flux lines and certain gauge theories that are similar enough to QCD that one can get a handle on how confinement qualitatively happens.
So the basic answer to your question is "no". It is as impossible for quarks to be wrong as it is for there to be no such thing as an antiproton. The evidence for quarks now comes from heavy-quark physics, where we can see spectroscopically the heavy quarks bound in non-relativistic bound-states with other heavy quarks. These charm-charm, bottom-bottom systems behave in just the way expected from nonrelativistic particles bound by a gauge force.
There is separate routine evidence from high-energy inclusive scattering. When you smash protons, you see jets, which are showers of particles in certain directions, and the jet emissions are correlated, so if you see three jets emerging, you can figure out the momentum of the objects which came from the collision point, and the probability distribution of the jet energy and angle can be calculated from QCD. The QCD calculations are in complete accord with experimental data, so much so that one has to go to high order of peturbations to match the distributions in complex multi TeV-energy scattering.
A: This question can be answered by analogies:
Thermodynamics was the pinnacle of the mathematical formulation of physics for a long time. Then statistical mechanics was experimentally discovered to hold, due to the particulate nature of matter with a whole new tool box for calculations.
Did this make thermodynamics "false"?
The quark model and its theoretical formulation is based on solid experimental evidence, as Ron Maimon points out in his answer. If, and it is a big if by itself, an underlying theory is ever found which will be necessary to explain new, unknown to us now, data, which may change the way physicists explain the Standard Model accumulation of data, the quark model will still hold, in a similar manner as thermodynamics still holds, though it is emergent from quantum statistical mechanics.
So no, the quark model cannot turn out to be false.
A: A point of view that is missing from all the answers so far is that particles are best viewed as excitations above some background (usually called the "vacuum"). This means that the particle content depends on the phase (or "material") that you're studying. For example, in a crystalline solid, you don't "see" atoms, but only vibrations of the entire crystal lattice --- a picture which becomes pretty useless once that crystal melts.
In the low-temperature, low-pressure phase that the majority of the observed universe seems to be in, quarks are confined and do not appear as the elementary excitations. However, at high temperature and pressure quarks and even gluons can become deconfined, to the point where they can be regarded as elementary excitations (this is still being probed and the consensus view on this is evolving). Incidentally, an amusing (but astrophysically realistic) case is high-density but low-temperature; in that case the current (as far as I know) view is that one gets a colour superconductor. Indeed, that phase may be the "same" as the normal phase (in that there exist smooth paths which connect them), and leads to the identification of baryons as quarks and mesons as gluons, appropriately dressed by screening charges.
From a theory side, they exist in so far as they are the fundamental fields used in the description, and that description has been shown to be remarkably good in a wide range of situations. Maybe one day we will discover substructure or something, but that theory must reduce to QCD in the appropriate limits, and in those limits we will still be talking about quarks and gluons.
A: OK, I think, you are completelly right with your lawful doubds.
We had seen that kind of approach in Medial Years. That time they told us that it is possible/impossible to count the number of angels on the top of the needle. Angel/demon confinment? :))))
The truth is that Chadwick in 1932-1934, knowing no negative patricles except from electron told them that neutron is elementary particle like proton with no real proof. Just (speculative!) statement. Sakawa, then Gell-Mann thusted that very and as a result we have quark model with fractional charges.
In fact, appart from that strange beliefs about neutron, we have to confess that neutron is composite particle consisting of proton and negative pion, so idea of fractional charges of quarks is just an error.
No fractional chages is possible in the Nature. Limited is our understanding.
A: The idea of quarks wouldn't ever be false, but the categories of the Standard Model are based on a certain unspoken assumption: that the physical ontology stays constant at such a small scale. 
