Obviously whoever built this was not very good at what they were doing and/or not really paying much attention to detail. When working with security critical things like authentication, that's not great.
Think of it like this: If you found that your electricity guy had accidentally installed the outlet sockets upside down, that in itself isn't the end of the world. But wouldn't it make you a little bit worried that he messed up something else more critical and your house will burn down?
Also, short limits to password length might be an indicator of a legasy system with no or bad hashing. Even worse, as jrtapsell highlights, the fact that the password is accepted when set but not at logg in could be a sign that it is truncated when inserted into the database and hence stored in plaintext.
So while this is proof of nothing, it should still set off your alarm bells.