While Jupiter-family comets are officially defined by (2< TJupiter <3), they can also be loosely defined by any comet with a period of less than 20 years, a relatively low inclination, and an orbit coinciding loosely with that of Jupiter's.
One such Jupiter-family comet, Comet Shoemaker-Levy 9, approached close enough to Jupiter sometime between the late 1960s and early 1970s, and was caught into orbit of it.