I have Photoshop Elements 11 and just bought Photoshop CS5. Should I uninstall the old one before I install the new one?
Photoshop Elements, despite its name, is not Photoshop.
Photoshop Elements 11 and Photoshop CS5 have nothing to do with each other, so you cannot call Photoshop Elements 11 "the old one" and CS5 "the new one".
They will not interfere with each other.
Unlike Elements, Photoshop is a professional level application that makes no apologies for its long and steep learning curve. If you are used to Elements, you probably want to keep it until you decide whether the full version of Photoshop is suitable for you.
One caveat: Adobe hasn't been selling CS5 for well over a year. Make sure you bought a legitimate copy you can run legally, as it absolutely requires not just registration but activation.
If you bought CS5 from a private party who has used its serial number to upgrade (to CS6 or to CC) then you would have an illegal copy of Photoshop that you cannot run.