OoP- Head Boy

[Naama]

No: HPFGUIDX 70149

In PS Hagrid tells Harry that James and Lily were "Head Boy and Girl 
in their day" (the Hut-on-the-Rock scene, I think). 

However, in OoP Sirius tells us James wasn't a Prefect. Lupin was. 
Then how could James have possibly been Head Boy? Mrs Weasley states 
clearly in OoP that becoming a Prefect is the first step to becoming 
Head Boy (she tells Ron so).

Could Lupin and (late) Sirius have lied about James so as not to 
make Harry feel worse about not becoming a Prefect? I can't see why 
they would. On the other hand, I can't see why Hagrid would- he was 
in Hogwarts when Harry's parents were at school (he's been there 
ever since becoming gamekeeper, hasn't he?)- if he told Harry his 
parents were Head Boy and Girl, it must be true.

Or mustn't it?!

Can you become Head Boy without having been a Prefect?