Yes, this can happen on all aircraft.
The static ports on the aircraft were covered with tape for cleaning, and then not removed. This resulted in contradictory flight data (mainly airspeed and altitude) being fed to the pilots, including that relayed from ATC. Similar actions in modern aircraft would have a similar effect. More modern aircraft have specially designed covers that are used instead of tape, making it less likely they will be left on.
In theory the presence of a measure of speed that did not rely on pressure sensors (such as INS or GPS) might provide some degree of protection, as might an altitude measure that did not rely on the plane's sensors, but it is not guaranteed even with those that the pilots could have figured out how to get reliable airspeed or altitude figures in the circumstances.