People are, or at least were, wrongfully convicted All. The. Time. Often - maybe even usually - the conviction is based as much or more on Being Different - visible differences like race, or social ones like not speaking the local language - as on actual evidence.
I'm sure the sentiment that
He's not One Of Us; he must be guilty happens no less now than it did in the past, but stricter limits on abuses by police and prosecution - including the strict requirements for the reading of rights, which came into effect a decade or two
after this story is set - improved forensic science (as stePH mentioned) and other changes have made wrongful convictions much less frequent and made overturning of them much more common. There have been dozens of wrongful convictions overturned in the US in the last 5 years, and those are just the ones that have been publicized.
In the 1930s, if a police force and prosecutor wanted someone to be found guilty and he (or she) was a poor, non-English-speaking immigrant with no one to stand by him but his (also non-English) family, he would almost certainly be convicted, guilty or not.
And even if the problem had not been rampant, it would only take one instances to make this story possible (well, that aspect of it, not the travelling house
).