I'm in the process of building a Drupal 6 site and I need to import
around 5,500 records from a Filemaker database into CCK. I've
exported a CVS file and I've tried the "node_import" module but I'm
having no luck whatsoever as it's still a work in progress.

I keep getting a dropped server connection or a blank screen when
moving from step 3 to 4.

node_import is the best solution for this.

I think you need to break/cut/split your huge cvs files into smaller files and try.