I am wondering if the order of your line segments is dependent on the order of your points. For instance, if your points were input in the order of 1,2,3,4 but lay along the line in a different order such as 1,3,4,2 - this might have some effect on how the tool operates because my guess is that it is going to proceed through your points as it encounters them (which is dependent on the order they were created in), not as they are ordered along your line. If that is the case, then possibly you can make this work more smoothly by ensuring that your points are ordered spatially along your line before you run the Split Line at Points tool.
One question I have is if you the segments you create are always the same length or if the segment length is always 1/5 of the total? There are tools in arcmap for splitting a line into equal length segments and for creating points along a line. But unless you start developing with arcobjects I suspect that you will have to do this manually, line by line, as I don't think there are equivalent tools available in model builder or which you can access from arcpy.