Except maybe the US flag in the US (given the particular US prohibition on copyright of federal government works), I don't see why not; and, actually, if the US were to adopt a private design as the official national flag rather than commission it, I'm not sure it would be a problem even in the US, since other private works adopted as national standards but not created by government in the US remain protected by copyright.
There are certainly ways a private copyright on a national flag is inconvenient, but inconvenient and impossible aren't the same thing.
"I, PHILIP MICHAEL JEFFERY, Governor‑General of the Commonwealth of Australia, acting with the advice of the Federal Executive Council and noting the fact that the flag reproduced in Schedule 1 and described in Schedule 2 is recognised as the flag of the Aboriginal peoples of Australia and a flag of significance to the Australian nation generally, appoint that flag, under section 5 of the Flags Act 1953, to be the flag of the Aboriginal peoples of Australia and to be known as the Australian Aboriginal Flag with effect from 1 January 2008."