RC522 RFID reader doesn't read NTAG216 chip
I got my RFID kit (RF522) just this week, but I haven't had a chance to play with it yet - nor do I have an NTAG216 implant. But I remember when I was researching them seeing a couple of posts that had to do with "Authentication errors". The problem for you (maybe) is that they were in Python rather than C:

https://github.com/mxgxw/MFRC522-python/issues/24

and Cotix's link

https://github.com/mj3052/MFRC522-Pi

Give them a lookover and see what they can tell you.

The problem with the PICC_IsNewCardPresent() function is that it gives a Yes/No answer, not a "Yes, but there's an error". In that case it would also return false. But the other code may give you a clue - you'll have to delve into the library...