Version of InDesign CC? If CC 2015, did you do the update last week to 2015.1? One of the problems it solved for some was saving issues.
Sometimes it's also an issue of not having permissions where you are saving.
Thanks so much Steve. I did update. But the wheel starts spinning every time I attempt to click the "File" tab. Yes, it is CC, btw.
Try setting the number of recent files to display to 0, then restart ID.