Can you be told you're trespassing in an apartment complex when invited by a tenant?

The easy answer is no, as you were invited. The problem here is that an apartment complex has what are known as common areas that are accessible to all tenants. If you were "caught" trespassing in a common area by the complex management or their security, it can be said that you are trespassing if you were not in fact invited by ALL tenants of the complex, which if course would be highly unlikely.

Assuming you can prove that you were invited however, this should not be an issue.

The fact that this question was asked begs the further question as to why the management/security felt you was trespassing in the first place. There are always extenuating circumstances.

Added: ALSO - if you have previously been "barred" from the grounds by the complex management you can be prosecuted for entering upon the property REGARDLESS of whether you were invited by an individual resident/tenant or not.