If the automatic transmission has been rebuilt properly then the rebuilt transmission should be just as reliable as a brand new transmission would be.
A professional transmission rebuilding place or transmission shop can put new parts inside the transmission and have the transmission perform just like new.
I've had a rebuilt transmission last over 500,000 miles and it's still going strong as long as you keep the transmission fluid and transmission filter changed every 30,000 to 60,000 miles.
You can buy already rebuilt or re manufactured transmissions and they will be just as reliable as the new transmission or rebuilt transmission you have a transmission shop rebuild.
If you rebuild the transmission yourself then it might be reliable but you must know where every little part and bands go.
Some people can rebuild them while others cannot.
I thought about doing the rebuild of the transmission myself but couldn't do so as it seemed so complicated but a good transmission shop can rebuild a transmission in one or three days if they have all the parts there.