Yes, although they will ~usually~ not mutually grant powers.
There are a few scenarios:
Character is devoted to Deity A (Let's say, Oghma, god of wisdom). Also due to similarities, he reveres Deneir as well (God os scribes). If you actually look at cleric interpretation, there are several daily duties and chants and prayers that are done daily - Only may believe and follow a second god's tenet, without drawing power from them.
Character is devoted to Deity A, whichever it is. Turns out character is also chosen of a different god, that may or not align with devoted deity. This often also leads to fun stories. Any power from chosen deity may or not mix itself with spell list, but will USUALLY come in the shape
of abilities rather than ordinary spells.
Character is devoted to Deity A, but dutifully is ALSO devoted to deity B. As of 3.5e, exalted deeds somewhere stated that multiple deities were possible, but since both faiths have in essence distinct in the ways of drawing power, and rituals, both levels do NOT stack, unless explicitly stated. Evidently this is up to the DM, but it would count as a dual class in which the character has two sheets for spell circles, like if it was a completely different class.
Character was devoted to deity A, but due to whichever inner conflict, deity was abandoned, or the true devotion was lost at some point. Still sympathizing or not with the god, the character will lose all powers previously granted by that god. There are two ways this is recovered - A penance to deity A, or a truthful conversion to deity B, in which the deity B accepts the strength and capabilities of the character, granting him power equivalent to that he is able to bear.
The reason why deities usually "battle" for devotes is because they also draw power from them. Their power and rank as deities are drawn from mortal devotes, who chant and keep this deity stable. The deity, in return, grants powers to those in favor, be they clerics, paladins, or any other divine based class or even item.
There are some very interesting settings about deities fighting over one or another mortal, whose mortal power is precious to them. (A.k.a. epic priests and stuff)
In some instances, deities even come to an alliance and "Share" mortals. Those are usually same-aligned deities. The other reply tells of the Triad, which is a HUGE thing in Forgotten Realms lore.
Forgotten realms itself has a few other cases an i can think of, either of alliances (Mystra-Azuth) and fights over mortal (Lolth-Mielikki).
As always, it's all up to the DM, but given how gods work in D&D lore, these conflicts are certainly a great spice to any story.