Atoms on the periodic table are defined by the number of protons in the nucleus of the atom. It is possible to discover/create new atoms by adding new protons to a nucleus, but those large atoms tend to be very unstable (which is why we don't add them to the periodic table anymore). There is a theoretical "island of stability" when atoms get large enough, but we are nowhere near being able to create atoms that large to test.

It is possible to create new compounds or molecules from existing atoms that we have already discovered, and those new molecules could have interesting new properties.