Is there a depth at which a human body, with lungs full of air, will no longer float?

The diver would cease to be buoyant when the their average density matches that of the surrounding water; where density is weight (or mass) divided by volume. The density of the surrounding water is not going to change much with such depths as water is very hard to compress. However the diver's density will increase as the air in their lungs compresses; their weight remains the same but the volume displaced by their body reduces and thus the average density increases.

The human body's density is relatively close to that of water; a common figure is 985g/liter, vs 1000 g/liter for fresh water and 1020 g/liter for sea water. That's only 3.5% less dense than sea water, so if compressing air in their lungs can reduce the overall body volume by 3.5% they will achieve neutral density.

Imagine an 80 Kg adult male (176 pounds). At 985g/l, they would displace about 81.2 liters. 80 Kg of seawater would displace about 78.4 liters (at 1020g/l). If the diver's volume could be reduced by 2.8 liters, they would reach equivalent densities.

A typical adult male's lungs hold about 6 liters of air; if their lungs could be compressed by 50%, ie: down to 3 liters of volume, their body volume would drop to 78.2 liters, and they would be slightly denser than sea water. Air in a balloon would be reduced in volume by 50% at a depth of about 30 feet (each 30 feet of depth increases pressure by about 1 atmosphere) - but of course the ribs and lungs are not a simple balloon and the lungs would not compress that easily but they go much deeper.

Overall, these simple calculations suggest that free divers might approach or exceed neutral buoyancy due to lung compression; at the least, their buoyancy would be significantly reduced.

Complications: body densities vary; they start with full lungs which reduces their initial density (but there is more than enough pressure to compensate for that); good freedivers have larger lung capacity; any gear worn would have some effect on the calculations unless it was neutral buoyant itself, and fresh water would make neutral buoyancy much easier to achieve.

Nevertheless it's quite credible that deep freedivers would lose their buoyancy.

Others have explained the physics. I just want to add that those of us with low body fat can become negatively buoyant at the surface if we breath out and empty our lungs. I can sink down and sit on the bottom of a swimming pool. So it is very possible to dive to a depth where you have negative buoyancy but what depth that will be will depend on the density of your body, or how fat you are (since fat is relatively low density compared to other body materials).

I have also personally talked to a free diver who confirmed that at a relatively shallow depth they become at least neutrally buoyant. This typically happens after they equalize the pressure in their ears, a process which uses some of the air in your lungs (if you use the most common shallow-depth equalization technique). If you have ever dived deep enough for your ears to start hurting from the pressure you've gone deeper than the point where you should have equalized the pressure. I think this depth is somewhere in the range of 15 feet. It's quite easy for even a relatively inexperienced snorkeler to reach this depth.

Realistically? No.

For this to happen, your body's density would need to increase faster than the density of water increases as you descend. This does happen because the bulk modulus of water is extremely high (about $2.2 \times 10^9\, \mathrm{Pa}$ compared to that of air (about $1.4 \times 10^5\, \mathrm{Pa}$). So ignoring the human body part of the problem, you could definitely compress the air in the lungs to a small enough volume that you're no longer buoyant. If you make some simple assumptions about how buoyant you are and your lung capacity, the math for figuring out the pressure (and therefor depth) needed should be simple.

The main problem is that compressing the air in your lungs is that your rib cage isn't a balloon that can expand or contract. There is definitely some minimum volume your rib cage can be compressed to before it collapses catastrophically.

The other problem I see is that compressing the air in your lungs will increase the temperature of the air quite a bit. Even if your ribs could expand and contract like a balloon you'd have to descend somewhat slowly to keep the temperature of the air in your lungs relatively constant.