Is it common to discuss the fact that one didn't have time to do more reading, or didn't have time to elaborate certain sections, in the "limitations" section of a thesis?
No. It not common, nor should you do it! As paul garrett pointed out, this type of excuse makes it seem as though you didn't care or couldn't be bothered to do a better job.
I would add also that no matter how much time and effort you have put into developing an idea and writing your paper, it is almost inevitable that there will be something that you missed or could have done better had you had/taken more time. Even the magnum opus on which you have spent your entire lifetime will be superseded by better or more complete thoughts; this is an inevitable reality of academic writing, and not one you should apologize for.
Perhaps especially as a student, one struggles to find the balance that represents maximum output for minimum time spent. Knowing when to 'let go' is a function of maturity and experience. When you've made that decision (or have had it made for you, due to time constraints outside your control), accept that this might be better, but don't apologize for the inevitable.
"No", for several reasons. First, even if literally true due to external constraints, it sounds too much like one really didn't care enough, or had other, more important things to do. Second, if it's a sort of excuse for not having a better paper, that's both unprofessional and will only make people mistrust you all the more. Third, although scientific and other scholarly literature has manifest limitations, to apparently concede huge limitations at the outset is pathetic.
Even if such remarks are a (misguided) attempt at some sort of modesty, don't do it. Your thesis is not "about you", but about its subject. Personal remarks should be limited to thank-yous and such in the acknowledgements.