Is it possible to have a convergent subsequence of a divergent sequence?

Sure. Consider $0, 1, 0, 1, 0, 1, \dots$

Furthermore, the Bolzano-Weierstrass Theorem says that every bounded sequence has a convergent subsequence.

It depends on your definition of divergence: If you mean non-convergent, then the answer is yes; If you mean that the sequence "goes to infinity", than the answer is no.