Is it normal for caulk to let go around my new home's tub and kitchen sink?
The answer is yes but with a caveat.


The caulk was not correct for the application.
Not enough caulk was used.
The caulk was an inferior make. 
The caulk was placed "outside" the gap, not binding in it.
Incredibly high variances of temperature or humidity.
The caulk was messed with.   When cleaning an area you cannot just scrub hard against caulk.   Eventually you will loosen even the best caulk job by pushing/scrubbing on it.


Each of these could easily cause an issue within a year.  If the builder used a good silicone caulk which should have been used in these areas and that said caulk failed in that short of time it is most certainly one of the issues above or a combination of them.