Well it sounds like your modem/router can't handle the traffic. How old is it?

Go to LW's prefs>Speed & tick disable Ultrapeer Cap. & press apply button.
Set both uplds & downlds to autoclear.
Close all LW windows you don't need open at that time.
If you're not in front of your comp, then hide LW in the dock.
Reduce uplds to about 70-80% or more. Reduce downlds to 80 or 90%.
Whilst LW is open, click on top of the Progress bar so all incomplete files are organised by % downlded. Then select 1st at 0% & scroll down to last & select it whilst holding down SHIFT key also all at 0% are selected, then pres the Stop button. Then go to your incomplete folder & view it in list mode & click on size column so all are arranged by size. Now delete all those at 0 KB. This will speed up opening of LW & reduce stress on connections & hopefully your router.
You might also consider reducing shared files if you have multiple thousands of them. Good luck. Sounds like your comp is losing communication with your router b/c it's too busy.

Also don't use the Resume button on all incomplete files. This will also stress LW & your comp. If you use Resume/Find sources then just apply it to a couple or so at one time.