My bike has disc brakes connected to the lever by cable. I've found I've been tightening the screw by the lever more and more to the point that there's not that much more room to tighten.
On most cable disc brake systems there's a moving pad and a fixed pad. The moving pad presses against the rotor and bends it slightly towards the fixed pad. The rotor is made of rather springy steel, and is probably where the feeling you experience comes from.
The barrel adjuster only addresses wear to the moving pad, but the fixed pad also wears in use. The rotor has to bend more to contact the worn fixed pad. This means that tightening up the fixed pad is necessary (about every other time I tighten the barrel adjuster on mine, but it depends on the pads and mechanism). It can be a bit of a pain as it's not very easy to get at.
There are two reasons why this happens with cable disc brakes.
The first is that the pads wear, and since cable systems are not self adjusting, you need to tighten the cable to bring the worn pads back closer to the disc.
Secondly, cables stretch over time, which again requires you to tighten the screw to take the slack out.
If you've reached the point where there is no adjustment left, then you need to check the brake pads don't need replacing. If the pads are good, then you'll need to un-clamp the cable at the caliper and re-tension it that way.