Searching for this issue online provides a lot of bad advice (one website I saw said “repair permissions and if that doesn’t work, reinstall the OS”… terrible advice.)
One solution to this is to reboot the Mac, but since Mac OS X is generally so stable and doesn’t need to be periodically rebooted as a matter of course, rebooting a machine with many open apps and windows can be a hassle. Instead, when you ‘paste’ in any app, the clipboard always pastes back the last thing you successfully copied.
Something has been bugging me with OS X for a while - sometimes the OS X clipboard (officially known as “pasteboard”) gets stuck and won’t accept any new ‘copied’ content.