Yeah that seems like coincidence to me... those are always scam popups in my experience.
Is your internet truly down or is it just that computer? If its just the one computer you may have gotten a piece of malware that is fucking with your machine.
Try another computer or phone or something and see if your internet works on another device.
If your internet really is down, I gotta think its a coincidence.... lol in college we got like 20 Copywright infringement warnings in one month on our Comcast account and they never shut our internet down. If there was a claim against you, it would be e-mailed to you on your ISP E-mail, you always have one with Comcast/ATT or whomever you use. Check it and see if there was even a claim filed with your ISP.
I seriously doubt they could shutdown your internet that quickly even if they wanted to rake you over the coals.... legal/corporate red tape is just too much of a bitch and normally they go after people providing pirate PPV streams, Ive not heard of them going after the viewers, too easy to spoof yourself and be a ghost for them to bother.
Check your internet on other devices and let me know.... I am betting its malware, easy enough to get rid of most of the time.