Impossible to offer much help without knowing:
What version of Word and OS?
Does this happen in all documents or just certain documents?
Does it happen with all hyperlinks?
How are you inserting the hyperlink?
Where is the hyperlink going to? same local hard drive, network? where
is the document?
Right-click a hyperlink that doesn't work and select Toggle Fields. Is
the hyperlink showing the correct address, or has it been changed? Try a
few hyperlinks, before and after saving the document.
(Note: if you have a single-button mouse, hold down control while
clicking to right-click)