Pressing the 'Check Mail' button doesn't seem to do anything.
When you press the 'Check Mail' button, any accounts that are currently in the IDLE state will be skipped (only IMAP accounts support the IDLE feature). If all your accounts are in this state, then nothing will appear to happen when you press the 'Check Mail' button (except the "## minutes remaining" resets). To verify that an account is working, you can right-click on an individual account and select "Check mail" -- this will force the account to refresh the message list and you should see activity as a result of this action; note that you can expect the following phases to be skipped: "connecting", "logging in" and "logging out".
The reason why POP Peeper skips accounts in the IDLE state is because they are already in a state where they will instantly receive new email notifications, and it would be inefficient to check these accounts. Some servers do not report new messages immediately to the client and, in these cases, it's possible that manually checking the account will reveal new messages that have been reported to the IDLE'ing client. These servers are usually on a set cycle when they will report new messages (e.g. once a minute).