Paddy is also a slang term for an Irishman, one that can give offense because of condescending, stereotypical associations. A police van, for example, is sometimes called a paddy wagon. The New Oxford American Dictionary speculates that that came about in the 1930s or so because many police officers in major Eastern cities were of Irish descent. I suspect that the term may be associated with the stereotype of an Irishman as someone who drinks up his weekly wages, becomes violent, and has to be carted away to jail to sleep it off. Your sense of the etymology of paddy wagon will depend on whether you think the term refers to the driver or the cargo. In any case, steer clear of it; you don't want to get anyone's Irish up.