Following the Create Fishnet tool help from scratch I didn't have the same issue in the same corner, rather my issue is that sections in Florida aren't necessarily square nor aligned to the reference system.
I believe the error you are trying to fix is caused by the use of the optional "Template Extent" instead of just putting the origin you want in the "origin coordinate" and the upper left coordinate in the "y-axis coordinate" input boxes. (Fishnets can easily be rotated to match imperfect north/south alignment and they can deal with rectangular sections that don't have perfect mile-long sides but they don't do so well on non-rectangular sections.)
The data frame is NAD_1983_StatePlane_Florida_West_FIPS_0902_Feet.
Origin= 606282.455 1235953.186,
Y-axis= 606282.455 1245953.186.
Cell size width/height = 1/8 mile
Rows and columns = 8
ArcGIS 10.2.2
The fishnet is 5280 feet by 5280 feet. In this case, the fishnet is oriented plumb with the spatial reference system but it can be easily rotated around the point of origin by changing the "y-axis coordinate" value. Because sections are not perfectly square ("sides intersect at 90 degrees" sense of the word) and the corners are not exactly 5280 feet apart from each other means that the fishnet won't really line up in any other corner of the section.
...
If you are ultimately after aliquot parts of sections (the square miles that PLSS is based on), you have to trust that the fishnet derived outcome is going to be "loose" and the error in both of our screen shots are probably not that far off from the other errors we will get when going down this path.
Maybe you could just use wider lines? ; )
There are probably as many reasons why sections are not square as there are efforts/methods to make aliquot parts with GIS. If you are not trying to make aliquot parts with fishnets, food for thought for readers that are thinking of that.