Oh, if only.
DIRECTV and DISH both sell over-the-air tuners for use with their own DVRs. Unfortunately they don't work with other DVRs, and the DVR that came from the cable company isn't going to allow you to get over-the-air signals.
If your cable company uses the CableCARD system for its DVRs (or if you have Verizon FIOS) you can use a Bolt DVR from TiVo
to combine both cable and over-the-air signals. Unfortunately this does mean an out-of-pocket expense as well as an additional monthly fee. There are other options that let you record antenna TV separately but then you don't see everything on the same playlist
Unfortunately, cable companies just don't take antenna reception seriously. It's a real shame actually since there is a lot of content they don't carry and that you can only find while using an antenna. But hey, that's why people get DIRECTV and DISH, right?