If there is a cap, regardless of how much it is, there needs to be a way for people to limit their ability to conceive - one that is accessible to women (or men) of any socioeconomic background. Accessible not solely as far as cost, but also as far as moral judgement. As long as birth control and abortion are stigmatized, then women will get pregnant above this 'cap', feel that they have no option other than to carry the child to term, and will therefore have to deal with the (far greater) costs of raising the child and having to deal with child care.
The lower on the socioeconomic scale you go, the less accessible contraception becomes - ergo, the more likely it is that people in that strata will have children (wanted and otherwise) above the arbitrary cap. It is far easier for a middle-class or higher individual to maintain their family size under the cap, since they have access to condoms, birth control pills, etc. With the religious right's vendetta against things like Planned Parenthood, the people with low income don't.
As a result, any cap without free and unstigmatized access to birth control for everyone is going to put the burden squarely on those who can't afford it. We've all seen how successful 'abstinence only' education is.