Most of the day I am connected to a lab or another workstation through Remote Desktop. Often times I need to copy (or cut) and paste from the remote desktop session. And sometimes this just straight out fails. When this happens, it’s usually just a simple matter of killing the rdpclip.exe process on the remote system and restarting it. If I am unable to copy (or cut) to the remote session, then I make sure to check that rdpclip process in running. If it is, then I need to close the session and reconnect to correct.
Often, the issue being encountered is caused by the so-called viewer chain. I found an old MSDN blog on the issue here: Why does my shared clipboard not work? (Part 1) & Why does my shared clipboard not work? (Part 2). Vista and up have done more to mitigate these issues.
As for why the clipboard stops working locally, it a shared service so probably some application has it opened and will not let go. In that case, your best bet is to start closing suspect applications until it works again (no straight forward ways I know of for identifying the guilty process).