Hello and welcome to HCM.
Thank you for your query.
Insertion of the ring can cause some menstrual irregularities, so the delay in period could be due to that. This was followed by the insertion of another ring, not giving your body a chance to go about the complete cycle which would have eventually resulted in a period.
Stacking of rings is not a problem, but you will have to experience your period at least once in a month or 2, or else this can lead to some repercussions.
Your question is quite good. Yes, in such situations, it is always advised to rule out
pregnancy, and so I would advise a home pregnancy test, using an early morning urine sample which would be most concentrated. Repeat the test every 5-7 days; totally conducting it 3 times. This will help in finding a pregnancy if present.
Once a pregnancy is ruled out, wait till you experience a period before inserting the next ring. If there is a further delay, do not hesitate to visit your doctor regarding the same.
I hope I have succeeded in providing the information you were looking for. Please feel free to write back to me for any further clarifications at: http://doctor.healthcaremagic.com/doctors/dr-shoaib-khan/64581 . I would gladly help you.
Best wishes.