No.  Counterexamples were first constructed by Winkelmann, as quotients of $\mathbb A^5$ by algebraic actions of $\mathbb G_{\text{a}}$.  I learned this from Hanspeter Kraft's very nice article available here: 

[Challenging problems on affine $n$-space](http://www.numdam.org/numdam-bin/item?id=SB_1994-1995__37__295_0).

Recently Aravind Asok and Brent Doran have been studying these kinds of examples in the setting of $\mathbb A^1$-homotopy theory, on the arxiv as [On unipotent quotients and some A^1-contractible smooth schemes](http://arxiv.org/abs/math/0703137).