Can a lizard shed a regrown tail again?
Short answer: many species can drop a regrown tail again and again again, however a few can only drop it once, while some lizards cannot drop their tails at all. It all depends on the species.

Background information:

According to this site the tail will only be able to be shed above the site it originally broke off. Although according to UCSB there are a few species of lizard that only have their tail grow back once.

Every site I've visited has said that the tail is nearly a completely different structure than the one it originally had. The original tail has "score marks" (fracture planes) which were studied by Sanggaard et al. (there are a few figures that you can see in their paper).

However, according to W. N. F. Woodland, D.Sc. (Lond.), there is no apparent limit to the amount of times a Gecko can drop its tail, "save the longevity of the animal". His paper goes into great detail, and has a lot of information about autotomy (limb dropping) of the gecko. I highly recommend going through it if you're interested. 

So tail re-dropping looks like it primarily depends on the lizard that you're studying.