Yes, it happens, especially with some RNA viruses. They are kind of “liquify”, change fast - and by the time our immune systems build up the antibodies against them, the manage to change - and slip away. A famous example is Hepatitis C. People recover, but they become what is called asymptomatic carriers. Actually, with Hep C it’s not problem free - those individuals actually continue to have a low grade active disease. There are good new treatments, but they cost a fortune.