Mine were born at 36 weeks (due to really high BP with me) and I have a close friend who had twins 6 weeks later at 34 weeks (her water just broke). Mine spent 9 days in the NICU and if I remember correctly hers spent almost a month in there but it was just a feeding/growing issue.
My experience with the NICU honestly wasn't bad. The hardest part for me was when they took DD1 to the NICU. DD2 went right away because she was only 4lbs 9oz and it was protocol for her to go and at least get checked out since she was under 5 pounds. DD1 went to our room with us and spent the night with us and on night 2 was taken to the NICU for a blood oxygen level issue. Between PP hormones, DD2 being in the NICU (and since she was in the NICU our hospital's policy was that only 4 predetermined family members would be allowed access so our siblings and other family and friends couldn't meet her), and not getting sleep I just lost it when they took DD1 away. But after that it really wasn't awful. While I was in the hospital myself it wasn't bad to just walk down to the NICU every few hours. The nurses there made me check my blood pressure (which was a concern) and looked out for me as well as the girls. As a FTM it was kind of nice to have medical professionals to ask questions to 24/7 for the first 9 days of their lives. When I was released we stayed at my parent's house because our place was an hour from where I gave birth but we would go to the NICU for their 9am feeding, hang out all day and just leave to get meals, then go home after their 9pm feeding. H also liked learning how to do everything with a nurse watching and giving him pointers. He left the hospital very confident that he knew how to do everything.
We slept there one night, but with the constant noises from their machines we got barely any rest. H was able to get a lot of rest at my parent's house, which definitely helped once we were home. We were also able to get them into a good schedule and routine while in the NICU, so when we went home with the babies we felt much more confident that we knew what we were doing then we would have if we had gone home at 4 days old. Mine were always just there for observation and help with feeding and growing so there were never any serious concerns, and because of that I was able to try to look at the positive side of things and tried to take advantage of the help we had available. The NICU was also brand new and really nice, and the girls were in a room together with a separate "bedroom" area at the back of their room with a sliding glass door and a futon and desk so we could go nap while they slept.
Oh, and the best part - they sent us home with a TON of stuff! All of the diapers that were left in the room, their diaper cream, tons of pumping parts and creams, thermometers, plastic basins, gauze pads, anything that wasn't nailed down they packed up for us.
Obviously the ideal would have been no NICU time and everyone's experience is different. We were very fortunate and there were never serious problems so we weren't too stressed, but we saw babies attached to all kinds of machines and I can only imagine what that must have been like for those parents. But normally from 34 weeks on babies do MUCH better. My OB told me afterwards that she didn't expect me to make it to 30 weeks, was shocked when I hit 34 weeks, and then was thrilled that I made it to 36 weeks (I was on bed rest from 26 weeks on for a shortened cervix). Just because they keep saying "34 weeks" doesn't mean that's when you'll deliver. That's just when everyone breathes a little easier. Hopefully you'll go even longer. Every day helps those babies. Good luck!
ETA - Other then still be petite (only 18lbs 5oz (27%) and 17lbs 10oz (17%) as of yesterday) my girls have had no other issues in the past year.