CAT-5 is what... 24 gauge? Check a table like this one, American Wire Gauge Chart and AWG Electrical Current Load Limits table with ampacities, wire sizes, skin depth frequencies and wire breaking strength. Now, which number to use... If it is a short wire, not in a bundle you can use the chasis wiring column. If the wire is in a bundle you will want to use the transmission number. If it is in a big bundle, even less.
As a sanity check, Power Over Ethernet moves 400mA, but that is over two wires and is spec'd to survive in bundles of 100 wires. 200mA/wire is well within the power transmission column.
In general, you will want to compute the resistance of your wire (remember to count the length of the wire coming and going) and decide if the power loss is tolerable.
Stranded wire holds up better. Flexing won't cause it to fracture like a solid wire. But that might not be problem in your application.