I think I missed the studs based on how easily the drill moved through the drywall. My LCD only weighs 27lbs and the mount itself is 5lbs (rated to hold 165lbs, and uses 4 8mm lag bolts). I pulled as hard as I could downward on the tv and nothing moved at all, everything seems pretty sturdy. Should I be worried? Is there something I can test in a week or so, to see if it will continue to hold strong?
Your TV is being supported by two pieces of paper. That should be all you need to know.
The white plaster material in the middle of drywall resists compression. The interaction between the shaft of the bolts and the inner plaster is keeping the load from shearing straight down the wall and will support a significant load.
However, on the front and back of the drywall are two sheets of paper that resists tension, similar to the purpose of rebar in concrete. And the interaction between the threads of the bolts and these two sheets of paper are all that are preventing your TV from being pulled out of the wall. Over time, the top bolts will pull out from vibration, temperature and humidity changes, and especially if you adjust the angle of the TV.
Once those top bolts start to pull through, the paper will tear, the plaster will shear out of the hole, and the TV will only be supported by the lower bolts under a highly leveraged load. Those lower bolts will immediately fail, and you'll have a broken TV on the floor at best, or a broken person that was hit by the TV at worst.
So, as everyone else has already said, remount the TV to proper structure.
Gypsum board (drywall, plasterboard, wallboard) is not a structural material, its only purposes are to prevent you from seeing into the next room and to conceal utility lines and structural members within the wall.
Drywall anchors are great for hanging small picture frames, little decorative shelves, and other similar knickknacks, they are not for hanging heavy expensive things like televisions.
When hanging any heavy object on a wall/ceiling, you should always anchor the object to structural members. If there are no structural members where you want the object, you'll have to make a brace or bracket that can attach to the structural members and attach the object to that. This brace/bracket can be either built into the wall (concealed), or attached to the outside of the wall (visible). In either case, it must be sturdily attached to the structural members of the wall/ceiling.
My advice to you would be to take the TV down, and remount it properly.
Did you drill pilot holes? If so, you usually get wood chips coming out and can tell if you've hit the studs.
If you just used the lag bolts to attach, I would be nervous about the TV. It partly depends on how bad it would be if it came loose. Is this a super-expensive TV that is right above your child's play area? Or is it just a small in a corner where nobody will bump it? You should also consider pull-out strength, not just straight down. E.g. if the corner of the TV catches on a passerby and they pull it away from the wall.
Personally I would remove the lag bolts and either move the mount onto studs if possible or use drywall anchors instead. (You maybe also should consider why you "missed the studs"—did you just drill and hope you'd get them? There are ways to tell where they are.) You can probably use the drywall anchors in the same location as the holes from the bolts.
Even though latest TVs are pretty light, you're not going to get insurance if you don't mount it on studs like you're supposed to. Use a stud finder and pre-drill pilot holes to be sure you got the studs. Your bracket will very likely have holes aligning at a regular stud width, so no need to pull off the drywall to install additional bracing. All you might end up with is a couple of tiny holes to patch up but you'll have peace of mind it will not come off by accident.
If you use some large toggle bolts and the TV isn't very heavy, you should be fine.
I built a wooden cabinet and filled it full of canned goods. I realized I couldn't get it where I wanted it while also having it on the studs, so I bought the biggest toggle bolts I could find and mounted it straight into the drywall with six of those bolts. It held just fine, even when filled with canned goods for a whole year.
So toggle bolts are always an option as long as the drywall is in good shape. There's also something called Snap Toggle that may be even better.