How does a prospective applicant fail to qualify for RAship, but fit the bill for TAship?
In the US, TAships and RAships are generally assigned by the department. There may be a rule about first year students (ours was that 1st year students could not be assigned TAships). You should also not assume that the division is made by a committee. RAships can be grant-related, or funded by generic department funds. The PI would usually make the decision if a student is considered especially appropriate for the project (unless the PI has a lousy department that doesn't give him/her much autonomy). Depending on the level of democratic sentiment, either the chair decides on the allocation, an admissions / funding committee does, or the whole department faculty does. TAships are also subject to university rules, if any, or department policy, about language competence, so it may be impossible to grant an incoming student a TAship if their English isn't certified for oral proficiency.

Generally speaking, an RAship is better than a TAship from the student's perspective, but the department might either have a particular need for a TA (prehaps in a specific area) in that year, or it might be advantageous to have a year's teaching out of the way so that in your second year you can do something else. I would say that you should not over-interpret either outcome, if you checked the "both" box. "Prefer" means prefer.