I had an oddball issue at a client recently that I hadn't seen before. After installing a fresh FAST for SharePoint 2010 server, configuring all of the services and provisioning a shiny new FAST Search Center, we were met with "Unable to display this Web Part" errors for all of the search web parts. Each time we refreshed the page one of the web parts came to life, so after 4-5 clicks of the F5 key, the search center performed normally. I searched and searched for possible causes, and the ULS logs were practically useless in tracking down a source of the problem. Then I came across this blog post which discusses a change Microsoft made to reduce the XsltTransformTimeOut property for the farm from a value of 2 to a value of 1. This essentially means that none of the XSL can take more than a second to transform, causing the web part to not be displayed. Changing this to a higher value gives the XSL longer to process and solved all of our problems.
$farm = Get-SPFarm
$farm.XsltTransformTimeOut = 3
Note that the reason Microsoft made the change was to reduce the risk of DoS attacks being performed through XSL. This change is likely not recommended for external or public facing environments.