I think that the phrase is intentionally ambiguous.
Originally 'by hand' meant nursed by someone other than ones biological mother (not necessarily by bottle). But Dickens is making a bit of use of amphiboly:
The full Dickens quote is:
My sister Mrs. Joe Gargery, was more than twenty years older than I, and had established a great reputation with herself and the neighbours because she had brought me up "by hand". Having at that time to find out for myself what the expression meant, and knowing her to have a hard and heavy hand, and to be much in the habit of laying it upon her husband as well as upon me, I supposed that Joe Gargery and I were both brought up by hand. (C. Dickens, Great Expectations
This reads to me that 'bring up by hand' mean something special and Dickens is making a play on words to the effect that 'by hand' evokes using her hand corporally against her husband and kids, so, ha ha, even though one of the kids was nursed by his mother, they were both brought up with the back of a hand applied.
Unfortunately, I can find no evidence of what I claim is the original definition in the OED.