Our last apartment had the same deal with the bathroom sink - looked OK
for about 3 months, then it started chipping. By the time we moved out
a year and a half later, most of the sink's paint was gone. I'd
definitely ask your landlord to redo it - shouldn't be your
responsibility. He or she clearly just did something as a quick/cheap
fix, and will probably do the same thing just before the apartment gets
shown to potential new tenants. Unless there's a real problem with the
finish underneath, or you're just bored and have always wanted to
repaint a bathtub every other month, I'd let it chip - it shouldn't
affect the performance of the tub.
Just my 2 cents.