We're contemplating doing this for dd2 (the other two have been to our local Catholic primary). I know some people feel that you can't do it for one child and not the other, but I think it all comes down to their individual needs, as well as the schools available (where we are, most secondaries are single-sex, for example, so the calculations are different for girls and for boys).
It is a balancing act, though -- I would for example like ds to have a bit more individual attention in his current class, and I know he'd love to be at the kind of prep school that does proper competitive sport, for example. But overall he's doing fine and has a lovely set of friends, so I think it's good enough for now. Dd2 otoh has an very low boredom threshold, is bright but with slight language delay, and is extremely wilful and quite naughty. It's an explosive combination, and I really don't think that the school my older two have been at is going to meet her needs.
I think it does depend on the degree of financial sacrifice that will be required of the family, though -- I wouldn't want to be in a position where family finances were constrained to the point that other children were restricted wrt eg. school trips abroad, holiday camps, music lessons etc. because of one child's need for private schooling.
As long as it won't affect other children apart from the issue of which school they go to, I think it's perfectly justifiable. To assume otherwise implies that the private option is always better for all children in the family, which I don't think is the case. We turned down a scholarship for dd1 to one of the very academic London independent schools in favour of her new state secondary without a backward glance.