What makes a good phylogenetic tree?
A 'good' phylogenetic tree would be completely refined, properly binary, have high bootstrap support for all its branches, and reconstruct clades we think exist with high support values.

Generally high support values(there are other methods besides bootstrapping but that is a question for another time) is a measure of a good tree. There are other concerns, for example a tree that has no felsenstein zones is probably better than one that has lots.

We can know a lot more about trees than you suppose, though. Through simulation studies we can create datasets with a known phylogenetic history and then see how well the methods reconstruct the history. There are still problems(errors in the model used, just random error, ILS, HGT, other things) but mostly that kind of study tells us what good phylogenies look like.

It's very easy to generate a lot of different phylogenies. Using different methods, different models or approximation schemes, different genes or sequences from each species, or simply bootstrapping "new" samples from your existing dataset. A phylogeneticist usually has dozens of trees to choose from, which is a sort of crude bootstrap method itself. If trees look similar, they're probably pretty good. If they don't, you have a problem.