Possible, but hugely expensive to practice.
Even if you have unlimited funds staying anon is becoming nearly impossible even without the complications of net surfing.
Its not really practical to travel with large sums of cash because the LEOs can confiscate it for no better reason then because they want to.(not to mention the risked of getting robbed by the amateur criminals rather than the pros in the Gub'ment LEO agencies.) So if you keep it in a bank, anyplace you go to withdraw cash will have a camera and more and more they are hooked to a system that has facial recognition software.
And once you get clocked into that system then if you go to surf on a public wifi more and more of these areas are peppered with surveillance cameras and of course most now are hooked to a computer system. NSA can poke around a bit and bang there is anonbob sitting with his laptop at the local coffee shop surfing the net. A little process of elimination of others logged onto the same node and well ANON no more.
Big Brother is real.