Check your External Testers list.
You can't have same email/tester exists in both Internal and External testers at the same time.
And, sometimes it does take a while to receive tester emails from Apple.

Just some minor observations that may help, (Nov 2015)

(*) after turning on internal testing, and/or adding an internal tester, it does take some time for that person to get the crazy "CompanyName has invited you to test AppName" email --- maybe an hour. This is ridiculous, but that seems to be how it is

(*) indeed note that there is a ridiculous "CompanyName has invited you to test AppName" email. Again this is for INTERNAL testing. you'd think that after you click internal testing, the app would of course just pop-up inside TestFlight. But no. nothing happens until the person (often yourself!) gets that email and confirms in the email

(*) that person (even yourself) has to use the hilarious Redeem Code feature, before it finally appears in TestFlight. Again this is just for the internal testing

What a mess!

December 2019 Solution

I got the same issue. Fixed:

1) Under the TestFlight tab, click App Store Connect Users.

2) Select Edit at the top right.

3) Try removing a user.

4) Click the + icon next to the word "Testers" at the top.

5) Add the person you just removed.

That should work and send them an email with a link to test.

It should also give you the option to resend invite to that person now. I think it's ridiculous that you have to do this, but it worked for me