I'd say if someone really wants to find out who you are and has the resources + time to do so then Bitcoin isn't anonymous at all.
If people know my address online, they know that I 'Chris!' own 1Chris4GEoLLjdh4juFXGwY7snaazuxvKb. If I start using transactions from this address they may be able to find out which company/person/entity owns that address and from there link it to me in real life.
It's so complex to stay completely anonymous with Bitcoin. The only real way is to buy Bitcoin with cash in the mail with no return address then never spend them.
Although at the same time, yes Bitcoin does provide a good amount of privacy because it's not like an email address. If you create an email address
joesmith1985@hotmail.com everyone knows your name and assumes your date of birth is 1985. This can be dangerous if you're hacked, if you post your email in a lot of places or if your name isn't very common!
All in all I'd say yes it's somewhat private but in no way is Bitcoin anonymous.