What happens if a small Tesla coil is fed with a powerful Neon-sign transformer?
The key to understanding is:  think about constant-voltage power supplies, where the load determines the current, which can be anything from zero to I(max.)   A very large AC line-transformer will happily power both the kilowatt bulbs and the milliwatt bulbs.
If your TC system is designed under the assumption that the HV supply is constant voltage 15 kVAC, then the physical size of the main TC coil is irrelevant ( ...well, as long as your air-gaps are wide enough, and will still prevent arcing during the peak volts: 1.414*rms.)  In that case a smaller Tesla Coil would just draw a lower current from your NST, compared to a larger Tesla Coil.   As with any transformer, extra iron and copper are required in your NST to avoid excess heating during maximum wattage, but this isn't a problem when your load draws less than the transformer's maximum watts/mA.
On the other hand, I remember old exotic designs where the NST and the main primary capacitor were critical for attaining max TC output (treating the NST secondary as a 60Hz or 120Hz resonant circuit, or, as a series load-inductance, where Vout is intentionally pulled down to less than 15 KVrms.)  In that case, the NST isn't just a constant-voltage power supply, so simply reducing the size of the TC coil might give unexpectedly poor results.  (But none of this matters for a standard, non-exotic TC circuit.)
Heh, if you have a huge NST, then think about ...christmas-tree lights!  Do something similar: try to power many little bitty Tesla coils, rather than one large coil.  I once made a Tesla-coil array with 80 TCs, each driving one fluorescent tube, as a &quot;Plasma sculpture&quot; at Seattle Folklife fest.
Also, years ago people were trying to achieve tiny TCs with yard-long lightning, by building them into the bottoms of tall plastic pipes full of oil!   Just how small can a TC-secondary be, while still putting out one entire megavolt?