Half my Irish family came in the 1850s. The other part came in the 1880s.
They were from County Mayo and County Galway (respectively): County Mayo was especially hard hit by the blight and famine. I don’t know for sure, but I’m guessing the County Mayo part of the family was escaping the famine.
I should clarify, the County Mayo part of the family show up in the 1850 census, which is why I suspect they were escaping the famine.