I have CAT5e jacks throughout the house and get gig speeds on every device.
Possibilities I can see... misbehaving NIC on the endpoint, damaged cabling, bent pins (or corrosion) on the jack, bad termination on the jack (or patch panel), misbehaving switchport, misconfigured NIC, could also be running CAT5 instead of CAT5e, amongst other things.
You can actually get up to 5 Gbit over CAT5e using mGig ports, so there shouldn't be any issues getting 1 Gbit speeds under normal circumstances. If you're not getting 1 Gbit over CAT5e, then there's an issue with something... somewhere... however subtle it may be. I feel like the CAT6 cable may be masking the underlying issue... but glad it's working.