Could neutrinos have more than 3 flavors? Since neutrino can change from one flavor to another periodically, given sufficient time could a new flavor emerges or is there a well tested rule that only permits 3 flavors for neutrino?
 A: Let's hypothesise a fourth generation of neutrinos: two opposite models have been considered.


*

*Active neutrino: that fourth neutrino is accompanied by a fourth generation lepton (the equivalent of the electron, the muon or the tau), forming a fourth doublet for the electroweak interaction; 

*Sterile neutrino: that fourth neutrino does not couple to any of the Standard Model particle, in any way.
In both cases, $\nu_e$, $\nu_\mu$ and $\nu_\tau$ mix with that fourth generation, which I will denote $\nu_4$ to avoid repeating myself, but the phenomenologies are completely different because an active $\nu_4$ can be produced in collider experiments whereas a sterile one cannot. Thus completely different experimental constraints exist for each hypothesis. The short answer is that active neutrinos must be heavier than about 40 GeV whereas there are a few hints that sterile neutrinos might exist, the current conclusion being that it is possible to have a sterile neutrino about 1 eV$^2$ heavier than the lightest of the 3 Standard Model neutrinos.
It should be noted that, in an earlier version of this answer, I only considered active neutrinos because I personally dislike the sterile neutrino hypothesis, which I see as the mother of all cop-out, and because I find the experimental evidences flimsy, but one does not dictate to Nature how it shall behave, hence this correction.
The long answers follows!
Active neutrinos
Elaborating on what I have just written, in $e^+e^-$ collisions at LEP, a $Z^0$ can decay into a pair neutrino-antineutrino, and that includes $\nu_4\bar{\nu}_4$ as well. Thus that opens two ways to find that fourth generation: $\nu_4$ could decay to a lepton of the first three generations, and that lepton would be detected; or it could escape detection and contribute to the so-called missing energy, to which all other neutrinos contribute too (neutrinos are never detected in collider experiments). The more missing energy, the larger the $Z^0$ decay width, which was measured with great precision at LEP, hence an additional experimental constraint.
So what were the results? The conclusion was that $\nu_4$ would be heavier than about 40 GeV (follow links in this Particle Data Group page).
Another point, because this is often misunderstood: if one assumes the Standard Model with 3 neutrino flavours but allow for them to be massive, then one can measure the number of neutrinos whose mass is less than half the mass of $Z^0$ (again LEP measurements). The best result is $2.984 \pm 0.008$, again from PDG. But that does not mean one has proven there are only 3 flavours, as discussed in the previous paragraph, since heavier neutrinos are possible. That result close to 3 basically just tells us that a fourth active neutrino has to be heavier, the lower bound being that given in my first paragraph.
Sterile neutrinos
Within this hypothesis, collider experiments are of no help since $\nu_4$ cannot be produced in the collisions of Standard Model particles. Only neutrino oscillation experiments can detect them, through additional disappearance of $\nu_e$ and $\nu_\mu$ than predicted with only the Standard Model three flavours (I do not include $\nu_\tau$ because we have just barely detected it in these experiments). Several neutrino experiments have seen slight deviations from 3-flavour models, which would favour a light sterile neutrino. Imho, none of those result is individually very compelling but taken together, they gain a bit of strength because they come from vastly different experiments, and most importantly experiments probing different neutrino energies. Circa 2013, two global analyses were performed, taking the data from all experiments and then trying to fit the same model to all of them. The result was that it is possible to have a sterile neutrino about 1 eV$^2$ heavier than the lightest of the 3 Standard Model neutrinos.
In the following, I will review each of those anomalous experimental results (when I say anomalous, I mean compared to 3 flavours). This is basically a pedestrian summary of the Particle Data Group review, section 14.13 where I explain how those experiments work in simple terms. It should be remembered that in particle physics, the standard for discovery is at least 5 $\sigma$, i.e. one chance in 1.7 million that the effect would be a fluke, and all the hints below fall short of that, being at or below the 3 $\sigma$ level, i.e. one chance in 370 that it is a fluke.
First we have the LSND and MiniBooNE experiments. They follow the same principle: a proton beam is dumped into a target, producing muons and anti-muons whose decays produce $\nu_\mu$ and $\bar{\nu}_\mu$. A detector is then placed in that neutrino beam at a distance of the order of 100 meters from the source. They both looked for $\bar{\nu}_\mu \to \bar{\nu}_e$ and $\nu_\mu\to\nu_e$ disappearances. LSND was the first experiment to find a slight excess for $\bar{\nu}_\mu \to \bar{\nu}_e$. MiniBooNE was then setup for a part to investigate that. The conclusion is that they saw excesses too:  in $\bar{\nu}_\mu \to \bar{\nu}_e$ disappearances at 2.8 $\sigma$, and in $\nu_\mu\to\nu_e$ disappearances at 3.4 $\sigma$. The former could be compatible with LSND but the latter is not, mostly coming from low energy neutrinos (less than 475 MeV). Most studies conclude that only the excess in in $\bar{\nu}_\mu \to \bar{\nu}_e$ should be considered.
Then we have the short baseline nuclear reactor experiments: they were performed in the 80's and the 90's in Grenoble (at ILL), Goesgen, Rovno, Krasnoyarsk, Bugey (so called 3 and 4) and Savannah Rivershow. They use the $\nu_e$ produced by the reactor and a detector placed a few tens meter from it (hence the short baseline moniker). The problem is that precisions experiments require a precise knowledge of the reactor fluxes of $\nu_e$. A re-evaluation of those was performed in 2011 and the subsequent re-evaluation of data showed an anomalous disappearance of $\nu_e$. But not everybody is convinced that the uncertainty on the fluxes is under control.
Finally, GALLEX and SAGE. Those are solar neutrino experiments using large vats filled with Gallium: a $\nu_e$ from the Sun may induce an inverse beta decay reaction where a nucleus of Gallium transforms into one of Germanium (an unstable isotope thereof). The amount of Germanium is then chemically extracted and counted thanks to the decay of Germanium nuclei. In order to calibrate them, some intense sources of ${}^{51}\text{Cr}$ and ${}^{37}\text{Ar}$ were used to calibrate them (those are radioactive by electron capture: $p+e^-\to n+\nu_e$ where the electron is from the lowest energy level of the atom). Those calibration data showed a deficit of the $\nu_e$ flux, at the level of about 3 $\sigma$, which can therefore be interpreted as an unaccounted disappearance of $\nu_e$. Since those artificial sources are very well understood, this seems difficult to explain.
