You can bring your Bible into China.
Where you might run into problems is bringing in a box full of Bibles, printed in Chinese, with a few thousand copies of religious leaflets.
Specifically, the law says that foreigners can bring in religious materials for personal use.
Seriously though, in the cities like Shanghai, Guangzhou, and Beijing, that have a large enough ‘laowai’ (foreigner) population to support a half decent foreign language bookstore, you can find three or four english translations on the shelf (I bought my KJV and my NJB in Shanghai, along with Dark Night of the Soul and a number of other specifically Christian books).
(EDIT: The only way I could possibly see a problem, and it still seems unlikely to me, but I should mention it, is if you are of Asian descent, so that the customs officials might consider you Chinese. Foreigner does not include ‘foreign born Chinese’. That is a special category. But if you are a blonde haired, blue eyed laowai, they would be more shocked to find out you weren’t Christian than you were. They think all European/N. American people are Christian.)
EDIT: My wife actually gets Hare Krishna and Jehovah’s Witnesses stuff in the mail here, and its very obvious from the envelope that they are religious materials. (She finds these groups fascinating for some reason.) However, neither of these are state recognized religions (the state recognized religions, I believe, are Catholicism, Protestantism, Islam, Judiasm, and Buddhism).