I’ve found that sometimes I’ve needed to toggle the ‘TestFlight Beta Testing’ switch to get everything to work.

UPDATE: iTunesConnect has been updated since the above answer. Now to reset it you should do the following steps:

1. Select the ‘Not Available for Testing’ then tap save.

2. Hit the ‘Select Version to Test’ on the right hand side. Select the version you want to test and then hit ‘OK’.

3. Finally hit save and then tap the ‘Start Testing’ button.
   

Now you should be good to go!

2018 UPDATE

Itunes connect seems to have changed again. The method above is no longer available. One solution us to expire the build, re-upload, and start testing again. To expire a build, select the build being tested (in the test flight section) the click expire in the top right corner. It should look something like this:

This seems to remove the build from TestFlight and you will need to upload a new build. Not optimal, but I can’t find another method right now.