The only time I could imagine that you'd need a specialist graph database would be if you had a very large number of nodes and some time / space intensive algorithm to run over them. Even in this case you could just store the data in a relational DB and use a low level interface (e.g. C) to write the specialist algorithm.
In any case putting the graph nodes and links into sql tables is a much easier to do option.
How does one know if it's time to migrate to a graph database?
Right now I have a large (6.5GB) database that I have to sometimes work with, and the queries are getting more complex as more features are added to the project.
Right now the data is stored in MySQL, but to get anything interesting out of it requires multiple queries and then sifting through the data later. It all seems cumbersome.
If you have a graph, then a graph database, with built-in graph algorithms, will be able to run operations on your graph without pulling all the data out to a client. I'm not an expert in PostgreSQL but I don't think it has any graph algorithms?