All three protons are equivalent, but they do not split each other due to quantum mechanical reasons.
Each of the methyl protons has two spin states available for something called population inversion. This is a quantum mechanical process by which a magnetic nuclide is promoted from a lower energy spin state (with the magnetic field) to a higher energy spin state (against the magnetic field).
However, the splitting patterns of any given proton is not due to the neighboring protons themselves, but rather the available spin states that the neighboring protons can have. Imagine a proton that is neighboring two identical methylene protons...
X-CH-CH2-Y
These two methylene protons can have different spin assignments (we'll denote a + for spin up proton and a - for spin down proton). Notice that "+-" is two possibilities, because either neighboring proton can have a spin up or spin down. These two assignments are identical in energy, hence their side-by-side placement.
++ (highest energy)
+- -+
-- (lowest energy)
This is known as a 1:2:1 triplet. Look at the ratios. One "++," two "+-," and one "--." Notice how many neighboring protons we had. Two protons therefore gives a triplet (called a 1:2:1 triplet because the triplet signal has one peak twice a intense as the others.)
In your case, the three identical methyl protons do not have any neighboring spin states available, so the signal would just be a singlet.
It gets very complicated in theory so we'll just leave it at that.