Short answer: Yes snakes have a fixed number of vertebrae from birth. Unless damaged (occasionally you'll find wild snakes missing parts of their tail from a narrow escape from a predator), they have the same number of vertebrae all their life. Different species vary in the number of vertebrae, and they can have hundreds.
This also has the effect of making them very variable in length (since each joint can move a small amount, the cumulative variability in snake length in very high), so they should be measured at least 3 times for a vaguely accurate estimate of length.
Incidentally, they also have the same number of scales in the same locations for their whole life (unless damaged)
This means that baby snakes feel incredibly smooth and silky like velvet. This also means that when telling different species of snakes apart that are otherwise very similar, "scale counts" are often used as a key diagnostic tool. Even very closely related species may have a different number of belly scales, or scales across their back, or different places on their head.
