In the late 19th Century, was it possible for a woman to divorce her husband yet?
Of course it will depend on which country you are referring to as to what divorce law was by say the 1880s and 1890s. In the United States, divorce was legal, and had been for some time, but not the notion of a "no-fault" divorce like we have today in many states. That is, you couldn't divorce for any old reason, a judge would have to decree the divorce on grounds he found suitable.
Needless to say, many courts in this country were not very favorable to women or women's rights, and certainly didn't believe in female equality. Drunkenness, rather than infidelity, was actually a more commonly granted cause of divorce, as alcoholism was a much larger social problem in those days, and temperance was gaining steam as a movement. A husband taking a mistress, on the other hand, was common and even socially acceptable in some circles.
In those days, it was also typical that the children would stay with the husband, as would much of the property and wealth. They made it as difficult as possible, in other words, for a woman to choose to leave a man.