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I've been told the courts don't deal with this sort of thing anymore
I don't know who told you this but the courts certainly do deal with it. Has there been an order of child support established? If so, was it established through CSA or through the Family Court? If it was established through CSA, it's true that they'll only take issue with the time they were involved onward. You can file a motion in Family Court to collect arrerages covering the period from the date of birth to when CSA got involved in the case. Of course, if the difference is not an inordinately long time (i.e. a week or two), it may not be worth the time and hassle going to court.