Continuity mistake: Early in the film Bud throws his wedding ring down the latrine and then retrieves it with his right hand so it is covered with the blue disinfectant. In a (much) later scene, Bud has a blue right hand, despite not having had one in the intervening period. (01:44:48)
Continuity mistake: When Bud is giving his wife CPR and starts to slap and yell at her you see her blink and swallow before they have any life readings. You also catch her closing her mouth as if to wet it when he bends over her body. (02:03:50)
Continuity mistake: When Bud leaves Jammer behind in the sub, he gives him a strand of rope to keep the two connected. At first Bud keeps the bundle of rope and hands Jammer the loose end, however in later scenes it's shown that Jammer holds the bundle of rope and Bud holds the loose end; as Jammer is feeding him line.
Continuity mistake: When Lindsay is pulled from the pool after she drowns her wet hair is covering her face. In the next shot she is laying on the floor and Bud is smoothing her hair back to give her CPR. The shot changes again and her hair is back covering her face like it was when they pulled her from the water.
Continuity mistake: When Bud sneaks on Coffey in the moon pool, they are seen fighting. During this Bud shirt gets wet and rides up his back. In some shots it goes down again, but then goes right back up.
Continuity mistake: When Linds says "I got it. It's headed straight for us," Bud looks out the front window. Watch the umbilical because it doesn't start to move until a split second after the camera is in full frame. (00:54:55)
Continuity mistake: When Jammer is alone in the derelict sub and the alien appears, all the electrical devices shut down, as they do in the whole movie. But in the close up shots from the side of Jammer, you can see the light at the top of his helmet is still on.
Continuity mistake: In the under sea duel in the mini subs Coffee smashes the tape player with his elbow, in the next scene, as seen from the outside looking into his sub, you can see it's still intact.
Continuity mistake: When Bud and Lindsay are in the sub after the big chase, water leaks in and Bud searches for something to fix it. He delivers some dialogue while opening (I believe) a first aid kit. The camera angle switches to Lindsay as she says something, and Bud can be seen in the background opening the kit once again. (01:54:44)
Continuity mistake: Bud swims to the moon pool and tries to sneak up on Coffey. Coffey aims a gun at Bud. Bud's right eye is swollen. In the next scene its not, and the following scene it is. (01:43:35)
Continuity mistake: When the SEALS take over, they lock up the crew. After closing the door on Ed Harris and the shouting Mary Elizabeth Mastrantonio, one of the SEALS hangs a hat over the spyhole. The next time we see the door, the spyhole isn't covered.
Continuity mistake: When Ed Harris gets used to the Liquid Breathing Suit you can see air still in his helmet after he goes in the water. That shouldn't be! The purpose of the suit is to get any remaining air out for it to work. (02:10:25)
Continuity mistake: When Lindsay turns to see the UFO, she has cables running around her, but in the next shot, no cables.
Continuity mistake: When the crane breaks off and falls into the water, it JUST misses the rig by a few feet (very visible when they're looking out of the window). Then the crane falls off of the edge and starts to pull the rig. The rig is pulled much farther than the previously shown distance to the edge.
Answer: The technology does exist, yes. Read http://en.wikipedia.org/wiki/Fluid_breathing for details.
Tailkinker ★