Certainly!  There are many who have used a menstrual cup successfully along with an IUD or Ring.
IUD
Inform your physician that you are planning on using a menstrual cup. They can trim the “strings” so that they’re not in the way of creating a good seal and so there is less of a chance of you tugging on them when removing the cup.
Make sure to release the suction before removing the cup to minimize the chance of dislodging the IUD.
Familiarize yourself with the length of the “strings”. Check them periodically to make sure that there is no change. If you feel like the strings are lower than normal, there is a chance that you may have dislodged your IUD. See a physician to adjust it.
Ring
Insert the Ring first and then the menstrual cup.
It’s common for the ring to move out of place while removing a menstrual cup. If this happens, insert a finger to move the Ring back into position.