Have we discovered more than 119 elements?
One of the first significant steps in understanding chemistry and the nature of matter was understanding the properties that defined an element. Once we learned that elements were based on the number of protons in a nucleus, we were able to predict the properties of elements that had not yet been found in nature. This is still the basis of elemental research today.
Nearly all of the elements lighter than 92 protons have been found in nature, so there is essentially no "discovery" left to be done. All new elements are created artificially in the laboratory, using powerful machines called particle accelerators. These machines can smash atoms together, forcing them to fuse into larger elements. Most of these elements are very unstable because of their size, and exist for only fractions of a second before falling apart again. It is only because of the advanced technology being used to create them that we are able to detect their brief existence at all.
To date, only elements up to 118 have been created in particle accelerators, but this information often changes as new experiments are conducted or old experiments are found to be in error. Hypothetically we may be able to continue creating new elements forever, but it's hard to say that they're really being "discovered" if we already know what they'll look like and we're creating them artificially.