Is it possible and safe to get both an upper and lower endoscopy done on the same visit?

Often it is: Depending on the indications for the procedures, age, other factors, both upper and lower endoscopies can be done on the same visit. Often the limiting factor is if insurance companies will pay for both the same day.

Yes: And this is often done by gastroenterologists, depending on the indications. Discuss this with your doctor to see if doing both is appropriate and necessary.

Yes and yes: It is not only possible and safe, but it is very common. It is actually more safe to do both procedures on the same day if they are necessary than it is to do them on separate days with two different sedation requirements. If they are both needed, you should go for it. Good luck.