Air carrier flight crew reported pressurization problem inflight. Returned to departure airport and landed uneventfully.

Narrative

We were in Aircraft X from ZZZ to ZZZ1. We had been plane swapped into this aircraft after our previous aircraft sustained damage during push back from the tug. This aircraft had an MEL for bleed 1. We followed the Mel and it was on the dispatch release. We took off and while climbing through approximately 14000 feet the anti ice kicked on due to icing conditions and everything was working as expected. We were getting a lot of step climbs and level offs as we were leaving the city area. At one point we leveled at 20;000 for a few minutes due to a lot of traffic. ATC then gave us a climb up to 34;000 but due to our MEL restriction of 31;000 maximum the First Officer requested 30;000 as our final. ATC then amended our climb to 30;000. As I went to initiate the climb we received a caution eicas message for bleed 2 fail. We asked ATC if we could remain at 20;000 to work the situation before going any higher which they said we could stay there for now. We did our non normal procedures and the first officer began working the QRH. I knew from this happening to me a few years ago that we more then likely were loosing pressurization. When I looked at the cabin altitude I saw it steadily climbing. I had mentioned this to the First Officer who stated that farther down in the QRH it said that if both bleeds were failed to descend to 15;000 or below and turn the apu on. Since we knew we were depressurizing we turned the apu on and I requested lower with ATC who immediately gave us lower. We used speed brakes and FLCH to assist with the descent and even with that the cabin got to approximately 10;000 feet and climbing. When we turned the bleed source back on from the QRH instructions the cabin started pressurizing again. We don't know if it was from the apu kicking in or the reset that was done via the QRH. At this point we were at 10;000 feet and everything was back to a normal" known state. We contacted dispatch via AIRNC to fill them in and speak to maintenance control. The decision was made to return to ZZZ between the two of us; dispatch and maintenance control. I notified the flight attendants letting them know we were heading back to ZZZ and that they had about 20 minutes to prepare the cabin for landing. I also made an announcement to the customers letting them know as well. On the way back to ZZZ we ran the diversion guide from the FCOM (Flight Crew Operations Manual) and also looked at the overweight landing checklist. We were showing we would only be 200 pounds over and now that we were back to a normal state we worked with ATC to burn the extra 200 pounds to be below max landing weight. We landed back in ZZZ and parked at the gate where passengers deplaned and maintenance control took the aircraft."