Continuing on with the concept of apophatic theology (referenced here), it's not something I've studied much, but it makes a certain amount of sense. If God is the infinite being that Christians believe in, then any labels we apply to him (including this pronoun) cause us to create a mental construct that's necessarily finite and therefore not an accurate picture of God.
Ultimately such an approach has its limitations; the biblical authors used positive terms to describe God on many occasions, which suggests that there is merit in doing so at least some of the time. Furthermore, if God is as relational as he (or she, or they, or no pronoun at all if you prefer) appears to be in the Bible, then we hamper our ability to have a relationship with our Creator if we completely abandon the use of positive definitions to help us in our relating.
Living in the tension between two conflicting truths can seem an impossibility at times: rely too heavily on positive terms to describe God, and before long God begins to look an awful lot like us, sharing our cultural biases, hating the people that we hate, endorsing the political causes we support and only loving those we view as 'sinners' conditionally. Move too far away from positive terms and God can quickly become an impersonal force in our minds, a distant power that we cannot relate to.
If there were a perfect balance that could be struck between the two extremes, no paradox would exist. But by living in the tension created by acknowledging both truths, we can self-correct as needed and continue to live in relationship with the God who defies all human definitions.