How would we know that Dirac equation does not describe composite spin-1/2 fermions? How do we know that the quanta of a quantized Dirac field describe elementary spin-1/2 fermions (or point particles of spin-1/2) and not composite spin-1/2 fermions (or extended structure of spin-1/2) such as a proton or a neutron?
Response to comment Surely Dirac equation doesn't describe the composite spin-1/2 particles. This is because, if it were so, the QED Feynman rule (derived assuming that the interacting charged fermions are described by Dirac theory) for the proton-proton-photon vertex in the case electron-proton scattering would be $-ie\gamma_\mu$. But it is not the case. 
Response to the answers I was reading Halzen and Martin. There they said that the vertex factor $ie\gamma^\mu$ cannot be used for protons because unlike electrons it is an extended structure. It is an effective vertex proton-photon vertex $ie \Gamma^\mu$ which contains the information that the proton is non-elementary. You may find the same thing written in the paragraph above equation (345), in the notes here.
But if I understand the existing answers correctly, they are suggesting that $ie\gamma^\mu$ is replaced by the effective vertex $ie\Gamma^\mu$ not because the proton is an extended object but because we are taking loop corrections into account.
I am confused now. What is the correct reason?
 A: The Dirac equation does describe composite spin-1/2 fermions - namely, baryons like the proton and the neutron. Conversely, future experiments might reveal the electron to be composite even though it's described by the Dirac equation (plus perturbative corrections).
The vertex term you describe does appear in the scattering cross-section for proton-photon scattering, but it's corrected by loop-level renormalization terms that stem from interactions, which are tiny (but measurable) for the electron but large for the proton.
A: Just as a complement to tparkers answer, people have been using the Dirac equation for composite particles since very long ago. Just remind yourself of the Yukawa model for hadron-hadron interactions
$$\mathcal L=\bar\psi(i\gamma^\mu\partial_\mu-M)\psi+\frac{1}{2}(\partial^\mu\phi)(\partial_\mu\phi)-\frac{1}{2}m^2\phi^2-ig\phi\bar\psi\gamma^5\psi$$
Note the first term, which is the Dirac equation for the nucleon in question. This theory is what gives us the attractive Yukawa potential
$$V(r)=-\frac{g^2}{4\pi}\frac{e^{-mr}}{r}$$
A: One succes of the Dirac equation is that it correctly implies the particle g-factor to be g=2, explaining the lepton g-factor. For protons and neutrons g is very different from 2, so the Dirac equation itself cannot be applied to these. 
The squared Dirac equation exhibits a spin dependent term, the relativistic generalization of the Pauli interaction. In this equation the g-factor of 2 can be substituted for by the g-factor of the proton or the neutron. In this case the modification takes account of the fact that these are composite particles. There are also loop corrections to the g-factor. These can also be taken into account in this way - but avoid double counting in perturbation theory. So the answer is: both.
A: Just to add to other answers and to bring light to the nomenclature : 
The effective vertex $ i e \Gamma^\mu $, that take into account loop corrections, can be written as follows : 
$$
\Gamma^\mu = A \gamma^\mu + B (p+p')^\mu + C (p-p')^\mu
$$
By using ward identity $q_\mu \mathcal{M}^\mu = 0$ we can get rid of the third term. Also, using Gordon Identity the second term can be exchanged for a $\gamma^\mu$ and a $\sigma^{\mu\nu}$ and finally be written down as : 
$$
\Gamma^\mu(q,q') = \gamma^\mu F_1(q^2) + \frac{i \sigma^{\mu\nu}q_\nu}{2m} F_2(q^2)
$$
It is these two terms $F_1(q^2)$ and $F_2(q^2)$ which give rise to the conflict here. These are termed as form factors. These can only be determined experimentally as far as I am aware. Form factors, historically, have had the meaning as corrections to the point-particle assumption. That is why Halzen-Martin say what they do. These form factors have come into play when calculating the loop corrections. 
Now, the chapter 3 of Halzen-Martin has very nice visualisation of interactions. When the interaction is happening at one point only the effective vertex is merely $\gamma^\mu$ (corresponding to the first term in perturbation expansion) and when you try to include more terms in the perturbation expansion (loop corrections) you need to have more than a point where the interaction takes place and those points are "spread over a region", hence the form factor nomenclature. Halzen-Martin include a nice diagram to illustrate the same.
Ref. for the effective vertex : Chapter 6 of Peskin-Schroeder
Ref. for the diagrams etc : Chapter 3 of Halzen-Martin
