My system won't hold a titer for mumps or measles even after booster vaccinations, though it does for ruebella. What might cause this?

An immune deficiency. Checking specific antibody production (or titer) after receiving a vaccination is how immunologists assess the function of your immune system. Even though your rubella response is okay, the lack of response to the others suggests an immune defect, especially in the context of recurrent infections or other findings like low immunoglobulins.
You're still immune. Measles & mumps are viral illnesses. The immune response is primarily mediated by t lymphocytes not antibodies made by b lymphocytes or plasma cells. There is an early antibody response to vaccination so you'll have IgG antibody titers for a year or more after vaccination but their subsequent absence does not mean you are no longer protected against these infections. Stop measuring titers & relax.