What kcchen said plus, you can also try scaling the print in all dimensions – this will naturally scale the gaps between parts. For example, I recently tried making this: 'Skeletonz' by muzz64 - Thingiverse
At 100%, some of the joints would move, but if I tried moving the stuck ones they just broke off.
110%: same result.
125%: worked great.
As you suspect, the issue is in part the nozzle width (but note the possibility of over-extruding). Much of the stuff on Thingiverse is optimized for a printer with a 0.35mm nozzle. You can replace the nozzle but you’ll want a torque wrench that reads 30 inch/pounds, and a 7mm crows foot adapter.
Thread about doing this: changing nozzle size
Lulzbot Instructions: https://ohai-kit.alephobjects.com/project/cf3b0638-1e1e-4bff-a00e-c3308d914c68/
I bought nozzles here, but it isn’t the only source: http://www.reprapdiscount.com/hotends/70-hexagon-nozzle.html#/nozzle_size-0_4mm/filament-3mm