It's quite tough to fool an expert, I'd even say that it's almost impossible. The problem is, if it's not a Picasso or Matisse, there won't be many experts that specialise on this particular artist, and no one wants to send a painting abroad just to check it's legitimacy (or pay a huge sum for an expert to fly in). Auction houses usually hire authenticity experts beforehand, but there are many cases when they were not ready to spend a huge amount on a real expert and went with a cheaper option.
It's almost impossible to pass chemical and physical tests that not only tell you the age of this painting but also, for example, the specific chemical formula of paint that the artist used at that period, but those tests are ridiculously expensive and they're never done unless there is a huge doubt over a very expensive piece's authenticity. Most of the fakes are easily spotted by a trained eye — it'll either be a random compilation of images from other works or a painting of a lesser-known artist from that same school/period with a little editing.
So it's all is about your resources. You won't find many fakes if you go through the best and the biggest museums that don't have problems with financing and employ the most competent personnel. But, obviously, the risk grows as the institution gets smaller. There is also a risk with international exhibitions — something that Ghent got caught in. They exhibited a private collection of Russian avant-garde artists and there were obviously no experts in Belgium on the matter. So, instead of going to the real experts, they decided to ignore the issue and the questions that were raised about the collection owner's credibility earlier. I suspect that the director was in on this though, and it was a corruption rather than an incompetence.
Most of the fakes go to private collections. Collectors are often more enthusiasts than experts themselves and there are tons of questionable influences around them that can offer you ill advises. Plus they don't often know who is the best expert on the particular artists, so they go to the first one, who may not be familiar with all the details needed for a thorough check. I know a lot of stories when the experts told the owner that their piece is fake and they did nothing about it — because that will only hurt them, so they kept pretending that they have an original piece.