It depends on your definition of quite common. If the accuracy of the tests are 98.8% then just over 1 from every hundred will get a false result. When you're carrying out 100k tests per day then yes, you'll get a lot of false results but as a percentage of the total it's very low.
When you're testing twice per week, as the PL and Championship are doing, the chances of the same person getting false negatives twice in a row is just over 1 in 10,000. Again, with the NHS carrying out 100k tests per day it's likely that you'll get some back to back false results but as a percentage it will be tiny. For the PL & Championship the likelihood of them getting 2 back to back false results is very small. And when you look at the likelihood of a PL footballer having 3 or more false negatives, you have more chance of being struck by lightening or Everton winning a trophy.
The above is no doubt part of the reason why the PL have began with non contact training. They'll know that some positive players have slipped through the net from the first round of testing and they'll know that there's a very small chance that somebody might slip through the net again in round 2. Nobody will slip through the net 4 times in a row, before contact training resumes, though.
edit: in responose to your edit, the accuracy of tests will be different depending on the tests being used. The tests the PL are using are said to be 98.8% accurate.