Double Secret Keepers?

[catherine at cator-manor.demon.co.uk]

No: HPFGUIDX 26786

Just listening to Fudge's account of what happened prior to the 
Potters being murdered, and I heard that Sirius Black was planning on 
going into hiding.  I thought struck me.  If Sirius was originally 
going to be their Secret Keeper, then why couldn't James or Lily be 
his Secret Keeper?  That way all of them would have been impossible 
to find.

Why didn't they think of that?

Catherine
(BTW:  I know the answer is that otherwise most of the plot in PoA 
would be redundant, but still...)