In practice, there are many variables to it. I know such theories have extended lists of ceteris paribus (with everything else held equal). Hold enough things equal and elasticity can always be higher in the long run. I'd like to add a few examples to the ones dsmithecon provided.
When people's buying power drops in the short term due to some crisis. One can switch from driving a car to a bicycle during tough times and then go back to driving the car when income goes back up.
New substitutes can become available in the short term. Same example, a bicycle would never be considered by some if they could afford a car. Historically this happened even to staples such as bread during famines when people would start eating stuff they never ate before. It was especially acute before central banks figured out to ease monetary policy during crises or there were no central banks at all. Or gold or strictly limited commodity or currency of another country was used as a currency, or still is in some places. Or there are constraining regulations in place. Or a command economy entirely.
I could probably extend this list even more, but the point is as far as I know there isn't an economic theory that exists without assumptions and can have a general proof or be completely refuted. If something is challenged, it is often added as an assumption and the theory may look sound again. It doesn't imply economic theories are wrong or even flawed. Economics, as all social sciences, deal with so many variables there is just no way to control them all and come up with something general, proven, and irrefutable.