There's always a discussion about this - all relative and good questions/concerns. Doing as much research as possible to where you are going is key. Bad things may still happen, but better preparation would help.
As a rule, don't carry all your cash and credit cards in one place. If you are able to take more than one card, maybe leave one in your hotel room and a bit of emergency Euro - €100. I understand that if people are on a budget trip, they may not have the "luxury" of having/taking multiple credit and/or debit cards. Some save for years to go on a trip once in a while. Do the best you can. Money belts or similar have proven to be very effective.
Being out of one's natural element can cause some havoc as LIZ in Pa pointed out. "Anyone" can get ripped off if the opportunity presents itself easily with a perp. In the USA, it is more customary to get mugged either forcefully or with a weapon pointed at you.
American travelers are sort of caught off guard about pick pockets because it is done so discreetly - in most cases. Then there are those who are too trusting or really not experienced with big city life and its adventures. Also, one is in "vacation mode." So, thieves scope the prospects and start to hone in.
It would be a PITA to lose all your money, cards and PP abroad. A local (just like we do in the States) can run to their own home bank, Dept. of Motor Vehicles, Consulate, whatever or whomever to get things replaced. A traveler or tourist would need to take a day or more to get things straightened out. And not knowing the language fluently would also be a hindrance. Also, your out-of-the-country bank would need a day or so to send you a new card or whatever. So, if you're moving a lot, as on a tour, that would be a hindrance.
Common sense, street smarts and awareness should still prevail. I'm sure, even at Disney World, people steal people's stuff - distracted by Goofy and Mickey - baby strollers with bags hanging off them. It's not right and thieves need to be dope slapped. But, if you don't take precautions to minimize or stay alert, then your stuff can be up for grabs for perps.