Does a new search kickstart the original download, or do you always end up having to start downloading a new file?

Are you resuming the download from the download pane or from the incomplete folder? Have you tried clearing the download from the download pane, then going library > incomplete files & resuming it from there? Does that make any difference?

Do you see the firewall icon (brick wall)? How many connection bars do you see?
What are your details?

OS(Win XP etc)
LW & Java version (help > about LW & you'll see)
Firewall you're using
Modem/router brand & model
ISP
Connection type (DSL etc)
Where you connect from (home, school etc)
Are you sharing a connection with other computers
Is this your first try with LW