Fluorescent lights will only work when it rains
Yes.  Unless there is a problem with the supply wiring or lampholders, ballasts are the only thing left!  

Look at the old ballast and see whether it brings 1 or 2 wires to each lamp end (in most cases, 2 wires).    If it's 2 per end, then look for a rapid-start (not instant-start) type ballast that supports the number of tubes you have, and their particular size.  Pay close attention to T12 vs T8 (different diameters) - an F40T12 and F32T8 tube will both fit in the same plugs - but require different ballasts! 

T8 is a bit more efficient than T12, but I wouldn't call it efficient enough to justify replacing a bunch of tubes you just bought. 

Cut the wires right at the old ballast.  Use blue wire nuts (the small ones) to splice the old ballast wires to the new.  

The new tubes are very good in terms of light.  To really see that, leave one corner lit with the old tubes, and don't look at the tubes, look at the stuff they're lighting up.  What a difference! 

Instead, you can "make the leap to LED" at this point.  They make LED tube-replacements for as little as $6, but quality matters - the cheapies don't have nearly as good light as true fluorescents. Whereas it's hard to buy a fluorescent these days that isn't 90 CRI. 

If you go LED, look for "ballast bypass" aka "direct wire" types - do not use plug-n-play types because they require a ballast, which is silly.  Also, seek ones where the 120V is fed from opposite ends of the tube, not on adjacent pins on the same end -- those little pins were not made to insulate 120V.