I'd actually argue it is possible to log too much if you aren't using good tools to make your logs easily searchable. Which is why you should use such tools if at all possible. Otherwise the logs can become so big that finding the entries for that bug or that email becomes pretty much impossible. This is also why it's important to take a few minutes and think about what you're logging and how. Things like request and user IDs can be invaluable. My test is usually "if I have nothing but this error message/log entry, do I have enough to begin investigating?". This is hard to get right until a bug or problem occurs and you actually need to use the logs but investing a bit of time into it can be a life saver.
