That's basically up to the Poles. It took me several months to get a bank in France to agree to let me open an account, including letters from my employer, landlord, proof that the landlord owned the property (he had just bought it) and so on. It's not easy in Germany; I know a nonresident who opened an account, but he needed a German friend to vouch for him. And in the U.K. (other than Jersey) I'm told you have to convince them that  you're not a terrorist.

All this may be different if you have sums in the 7 digits or up to invest.