We have a three server SharePoint 2013 farm, 2 WFEs 1 APP/CA server.
We were trying to get SQL reporting add on installed to one of the front ends. After the install it started behaving strangely, images were getting permissions issues, browsing to the root of a publishing site would prompt for an access request, etc. If we shut that server off it started working again.
We've tried reconfiguring the distributed cache... It seems to work for a few minutes and then starts failing again. Not sure where to look at this point.