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Although I agree that it sounds like a connection issue and not spyware, I would have to say in my experience spyware is not very well written and can have serious bugs that can screw up a system. So even though there's no purpose in spyware halting downloads, it may be doing it accidentally.
Are you running a hardware firewall on a router or software ones on the machines? If so, does it still happen when you turn them off?
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