We don't use tea bags however we don't have the 'dust' problem either. I have noticed though that some tea strainers are not as fine as others. If I'm using cheaper teas I use a finer tea strainer - sometimes even the plastic mesh one for 'kitchen' teas.
I do have some delicious 'special' teas, which are treats - and they tend to have much larger leaves.
No one in my family (to my knowledge) has ever deliberately left tea in the cup - but my grandmother did leave me her tea set when she died, which included a massive, matching, china slops bowl!