Can soy milk cause breakthrough bleeding while in birth control pills?

Not very likely: soy milk doesn't have a lot of chemicals/hormones/interactions with medications. Still, you might want to avoid having a dairy product in the stomach at the same time as a medication. OCP's absorb well, so this isn't very important as an interaction, but other meds such as tetracycline bind to dairy and don't get in.

Yes, possible: because large quantities of soy milk can influence estrogen. Moderation is the key.