It may vary depending on the TLD.
Any .uk domain (overseen by Nominet) requires correct contact information:
What you must do
3.1 You must:
give and keep us notified of your correct name, postal address, phone and email contact information. This includes responding quickly to any request from us to confirm or correct the information on the register
This means that the contact information you or your registrar provide us with must be good enough to allow us to contact you quickly at any reasonable time, must not be deceptive, and must clearly identify you
I image that other domain authorities have similar rules, as you are entering into a contract by registering a domain.
ICANN certainly seems to have similar requirements: https://www.icann.org/resources/pages/benefits-2013-09-16-en
Apart from running into difficulties if you have violated the terms of registration, you will also have problems if you ever need to prove that you are, in fact, the registrant. I'd strongly recommend using correct information and a privacy protection service, if you feel you need to.