Both years I interned the exact same thing happened to me. I thought getting crap done and messing around on the web was just an internship-type thing.
Year 1: Got my stuff done in decent time (wasn't trying to go fast), the job was going well and I had a ton of free time on my hands. Usually went home around 3:30 or 4 and they still allowed me to count a full day. Did this all summer and within a week or 2 after I got back to Manhattan (stud internship was in DC), I received an offer for an internship the following summer.
Year 2: Did not take internship from previous company so worked for a different company doing stud stuff in Florida. Got done with my work fast and efficiently so played games on the PC (while at work).