Like everywhere else the basic design principle of contrast also holds in the tone of error messages.
If you shout all the time, nobody will listen when you really mean it. Likewise, if you are sorry all the time, nobody will take your apology serious.
"I'm sorry" has two very different uses -- it can mean, roughly, either "I accept responsibility for this thing which is my fault" or "I regret this occurrence, independent of whose fault it is". When used in the former sense, it doesn't necessarily imply that there will be no recurrence, but certainly implies that the speaker will make an effort to prevent recurrence. When used in the latter sense -- which I'd argue would be the usual sense in which an error message would have to be understood -- it certainly does not imply even that there will be an effort to prevent recurrence.
If I see a one off apology, for a one off issue, perhaps with a mild sense of humour, then that's good, I like that. I might even have sympathy.
If, however, I see a regular, frequent issue and get the same apology over and over again, it begins to feel insincere, and that they are trying to cover an inherent issue with an insincere apology, or humour. I then begin to feel that they simply cant be bothered.
I think sincerity is the most important thing.