It is true!
I am a Licensed Massage Therapist as well. I agree with your LMT. If there is tightness in the tissues, pressure on the area -- sometimes even light pressure -- will be painful. In fact, that can be a good sign because then you will know that your therapist is in the right spot!
The good news is that as the therapist works the area and as the tissue starts to let go of the tension, the pain will minimize.
If you were to press very, very hard into tissue that has no tightness, you will feel it but it won't be painful -- if you press very lightly into tissue that has tightness, it can feel excruciating. The best cure -- deep tissue or a stable pressure, focused intention by the therapist and a desire to let go of the tightness by the client.
You should feel relieved that you are in good hands!