Is it common to experience knee pain and have the feeling that your knee is giving out, when you have patellar tendonitis?

Absolutely: Knees typically give out for 3 reasons: pain, muscle weakness and instability. The first 2 reasons are likely why you have this sensation.

Yes: When you have knee pain, especially involving your extensor mechanism (quad muscle, tendon, patella, patellar tendon) you don't fire your quad right and this can cause feelings if weakness or buckling. Rehabilitatin should help this.