NM wire melted in attic
They sell  "850 watt" power supplies for gaming PCs.  When you look at the power supply sticker (we're talking the responsibly built, UL listed ones), they indicate 10A / 1200VA.  How much power do you need to provision to such a machine? 1200VA, not 850VA.  The difference is conversion losses inside the power supply (which is why power supplies have fans), including power factor. 

So I'm a little skeptical of where this "216 watt" figure came from.  It is precisely a multiple of both 6 and 4, and "36 watt" and "54 watt" are standard ratings for fluorescent tubes.  However, extrapolating from nominal tube spec totally disregards: 


over-unity ballast factor (meaning the ballast drives the lamp harder than its nominal spec)
internal ballast inefficiencies and conversion losses
power factor of the ballast (VA vs watts; you must provision VA). 


I also don't put a lot of weight on your 18A (2A per fixture?  Exactly?) number.  I would be happier if you obtained a Kill-a-Watt meter and plugged one of the fixtures into it, and see how it actually performs at operating temperature.  Look at amps, and VA.  (Watts is useless). 

I think you will find you have significantly oversubscribed this circuit. 

The cable being under a carpet of insulation isn't helping either, but I'm not an expert on Romex installation rules, so I leave that to others. 

Continuous loads must be derated to 125%

Lighting is a continuous load.  So it must be derated: you must provision 125% of the VA that the lighting actually requires: NEC 215.2a1.  

Another way of looking at this is, if all the loads on a circuit are continuous loads, a circuit can only support 80% of breaker rating.  So a 20A circuit is good for 16A only.  (16A * 125% = 20A). 

Even with your perfect 216W number, your 9 lights would draw 16.2 amps (they don't) and that's a tick over limits.  In actuality I think you will find your lights are even more than 2.00 amps each, so over 18A requiring you provision over 22.5 amps of power to the circuit.   Upsizing to 10AWG wire would reduce your wire overheat, but you would not be allowed to upsize the breaker to 30A, because you are not allowed to put NEMA 5-15 or 5-20 receptacles on a 30A circuit (table 210.21B3).

Break it into 2 circuits

The most straightforward solution here is peel off 3-4 of these lights onto a separate circuit.  And, you know, if you want to be able to switch some of the lights separately from the others, this is a great time to do that. 

Or... switch to 240V

Pull down a light and read the labeling carefully.  Given these T5 lights are super popular in Europe, and given trends in the lighting industry to make multi-voltage ballasts that work on 100, 120, 208, 240 and 277 volts, chances are these luminaires are already capable of running at 240V. 

So convert the circuit to 240V. The switches won't care. The lamps won't care.  


The plugs and sockets must be changed from NEMA 5-15 to NEMA 6-15 or 6-20.  (they look practically the same and use the same cover plates). 
The lighting circuit must be punched down onto a 240V breaker 
White wires in conduit must be replaced with colored wires.  
White wires in cables must be taped black or any color at both ends. 


What will this do to wire heating?  It will cut current in half, which will cut voltage drop in half, which will reduce wire heating by 3/4.   it's a very good deal that should solve your overheat problem.  As long as you don't add even more lighting or other loads.