Yes you can desinstall and reinstall, then re-enter registration data if it work.
If still an issue, add "DebugSQL" as a parameter in the shortcut to PL/SQL Developer, run it, let it hold, and find the debug.txt file in the installation folder (or in Windows Roaming folder if you are in Windows 7+ and installed in Program files -- see documentation for location) and check if you find the reason in here, or send it to support for help.