I hate to bring math into it, but the ticket is probably valid. Let's say that you pay for 30 minutes of parking at 1:10:00pm. You then get a ticket at 1:40:01pm. You've stayed in the parking space for 30 minutes and 1 second. Your parking technically expires at 1:39:59.99999999999...., but just before 1:40:00.

Most likely, their automated system reads the actual time and not just minutes. It's possible that you were only 30 seconds over the time, but got the ticket and that truly sucks.

If the city doesn't overturn the ticket, you have basically zero chance before a judge. In order to appeal it to a judge, there is a non-refundable $275 filing fee. If you win the case, Cambridge has to forgive (or refund if you paid) the fine. You don't get the $275 back from the city or the court. So if you got a $50 ticket from the city, it'll cost you $225 if you win in court and $325 if you lose. It's a lot better to pay the $50 ticket. Yes, it is somewhat problematic that you can't actually appeal the tickets, but that is how things are at the moment.