Can I download an OSM Planet file with FTP?

If you use wget on the command line to download the file (e.g.: wget http://planet.openstreetmap.org/pbf/planet-latest.osm.pbf), and it gets interrupted, then you can just pass in the -c option and wget will automatically continue downloading the file. e.g.: wget -c http://planet.openstreetmap.org/pbf/planet-latest.osm.pbf. If that gets interrupted again, you can just call that command again and it'll pick up where it left of (etc. etc.)

It is not true that HTTP downloads cannot be resumed (see e..g the --continue option of the wget command line utility, or use some kind of download manager).

There is an FTP mirror of OpenStreetMap data, including old planet files, here: ftp://ftp.heanet.ie/pub/openstreetmap.org/

Note that if you download the "full history planet" then you can make snapshots for any point in time from that, and don't need to download tons of planet files.

Your assumption is not true. HTTP downloads can be resumed. This feature has to be supported by both the server and the client but it is quite unusual to not support it (except for many one-click hosters and similar services). Hence you should try to find out why you can't resume your download.

Resuming a HTTP download from http://planet.osm.org works fine here.