I watched a video of a dosbox game that was supposedly running at default cycles. However, it was noticeably faster than on my computer with default cycles. Can that happen, or must the other video have had cycles turned up?

Yes, that's normal. The "real" speed at n DOSBox cycles depends on your CPU.

EDIT: It depends on the game of course, but unfortunately most DOS games from that era followed the philosophy of "run it as fast as the PC allows it". Only minority games of games ensured that the speed would be constant regardless of CPU.

The person who made the video probably had already fired up the game and changed settings as well. I know some Let's Play folks do a test run video on anything new. That way they don't have to waste their time installing and setting things up in the real first video.

Well some do that. Some don't.....

I remember when Fleet Defender needed a patch for any machine modern enough to run Windows 98 because the faster computer speeds would make the old DOS game fly the plane at absurd speeds.