Ya cant use all of them, that should be the reason.
It depends on their level and your fishing proficiency.
As far as i know, you can only use splintered rods till proficiency lvl 5 or so.
Just drag and drop it into that box when you click on the fishing pond and you can use it.
Its WAY better to go with these instead of a loaner.
But be careful, if you loose the battle against the fish your rod will break and you cant use it anymore.
Rather release the line and run if it gets too close
Edit: Just checked and discovered a couple of extremely nice splintered rods for your fishing.
Dont risk too much with them, it would be a pity if they break
