I am not a rebreather diver. As such, my uninformed opinion is that rebreathers are unsafe at all depths. OK, that was a joke, but there is a small element of truth in it. Rebreathers can be used in shallow or deep water, but the benefit of being able to add O2 and scrub CO2 without wasting the gas which carries it (called diluent, sometimes nitrogen is used) really begins to shine at depth. Let's assume that you are at 120 feet. Every lung full of air you breathe in is really the same amount of gas as four lung fulls of air at the surface. This means also four times the amount of oxygen in your lungs at 120 feet as you would at the surface. With regular (open-circuit) scuba gear, when you exhale you are wasting a lot of good oxygen and lots of nitrogen, all to get rid of some CO2. Rebreathers scrub out the CO2, add a shot of O2 when required, and keep the diluent (nitrogen or otherwise) circulating. It is a very efficient system. The problem is that if something fails, you are likely to get air that is too rich in CO2, or too low in O2. Since you are still breathing you might not notice this before you pass out. A failure of open-circuit scuba often means that you are getting no air, and that you will notice right away.
FWIW I'd love to learn to dive a rebreather. I am waiting for the day when rebreathers are well-designed, compact, have multiple redundancies, and are not incredibly expensive.
Edit: At surface pressure, you breathe in air at 21% oxygen and exhale it at about 16% oxygen. There is clearly good O2 in the exhaled air - if there wasn't, mouth-to-mouth emergency breathing would not work. If you descend to 30 feet, you inhale the equivalent of two lungs full of air, still with a 21% O2 content. However, you exhale the air at about 18-19% O2 content since your oxygen requirements have not increased with depth. At 30 feet you are wasting even more O2 than at the surface. This waste gets worse the deeper you go - thus, rebreathers get "better" the deeper you go.