I'm running QGIS 2.8 Wien on Windows 8.1. I got this error:
The program can't start because qgis_core.dll is missing from your computer. Try reinstalling the program to fix this problem.
I got the same error last week, uninstalled and reinstalled QGIS and it worked. But there must be a better way to fix the error than reinstalling QGIS every time it bombs out.