That depends......If the supplier is supplying materials to a first tier sub, then yes it would fall back on the GC. It has to be a supplier that has a contract with a sub who has a
direct contract with the GC. That is a little confusing, so I have pasted a link to the SIO page on the Miller Act.
This is why it is usually best for the GC to bond all direct subs, that way the GC has recourse against the subs bonds if they do not pay their suppliers.
The Miller Act