I'm on international assignment and the length of my stay dictates that I am not eligible for a personal number. Without the personal number it is still possible to get a Swedish bank account (by law they actually have to give you one in Sweden as long as you know how to properly request for it) but as an someone hinted below they definitely do not make it easy for you to get one if you are American specifically or foreign in general. Even Danes that have come to work in Sweden have stories of how difficult it was for them to get Swedish bank accounts, having to quote the relevant part of the law that states that they must be awarded one upon request.

It also makes my taxes back in the US quite a bit simpler to state that I have no foreign account holdings whatsoever (I am still paid in my US bank account)

I cannot speak for the OP, and I don't live in Sweden, but as someone freely travelling and living in many EU countries (I'm European) I can tell you that opening a bank account as an expat has become seriously hellish sometime in the the past two or three years. Transfers over certain amounts have become difficult, and more often then not, my business transactions (client invoicing) are blocked arbitrarily, pending "evidence" - 2 months ago it took me 4 weeks, and thousands of euros in legal fees to release a legitimate payment from a client to me.

In the new EU, "your money" is becoming a myth. After the Cyprus Experiment, where the banks essentially took ownership of all money, they realised they could get away with it. KYC, AML, FACTA and related bullshit are seriously concerning me. Combined with the relentless and useless drive to this "cashless society" (why do we need this?) your money is not yours any more.