Until recently it was not possible to download more than a few hundred species at the same time. This is unfortunately is still true for downloads through the portal, which are limited to around 200 taxon keys (species, genus, family, kingdom …). This is due to limitations in browser and the http GET limit. A recent change, has now made it possible to request more species names (up to 9000) in some cases through an http request.
This is a companion discussion topic for the original entry at https://data-blog.gbif.org/post/downloading-long-species-lists-on-gbif/