I'd venture a guess to say that in other campaign settings, where the gods are less possessive and jealous, that might be possible. But in the FR, and especially the FR after the Time of Troubles, I'd imagine the gods are kind of persnickety about distributing their blessings to someone who couldn't care less about that specific god's dogma. (Which would be the case for someone who believes she's getting her spells from nobody in particular.) The gods are well aware that their own power is directly tied to the total strength of belief of their worshipers, so why would they squander that power on someone from whom they'd not be getting any overt service in return?
From an OOC standpoint, the FR is unlike some other campaign settings in that it's explicit in the rules that a cleric can't just give her faith to an entire general pantheon, and a ranger or druid can't just worship disembodied, non-personified nature. A divine caster *must* specify a single deity from whom she gets her powers, and that's not some subconscious thing that that you can sort of stumble into without knowing it, I think. It has to be an expressly willful act of devotion, a conscious decision on the part of the character.
Just my opinion, of course.