I've been asked this question by a colleague who's not an algebraic geometer; we both feel that the answer should be "no", but I don't have a clue how to prove it. Here's the question: let $X$ be a smooth rational variety (over the complex numbers, say). Is it true that every point of $X$ has a Zariski open neighbourhood that is isomorphic to an open subset of ${\mathbb P}^n$?

Some partial results related to this open problem can be found in the very recent preprint by F. Bogomolov and C. Böhning On uniformly rational varieties, see arXiv:1307.0102. According to the authors (see the Introduction) this question was first raised by M. Gromov in his paper Oka's principle for holomorphic sections of elliptic bundles, Journal of the American Mathematical Society 2, Vol. 4 (1989). 


Here is a preprint of Ilya Karzhemanov constructing counterexamples in dimensions $n\geq 4$: 

