It’s definitely possible to have multiple network connections to the same computer. But in order to make sense, they really need to be different networks. Otherwise you might not get the results you’re looking for.
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By different networks I mean … well, different networks. Different collections of machines.
Here’s what I mean: you current have two machines, A and B, that are connected to a wireless network. I presume that you can, for example, copy files between the two machines today. If you connect both of them to a wired network, in addition, then there are two paths to get between the machines: the wireless connection and the wired one.
How does Windows decide which to use? Well, while I’m sure there’s an algorithm behind it, the best way to think of it is that the first time the two machines try to talk to each other, the connection used is selected randomly – it could be on either network. From then on (I believe until both machines are rebooted, or their cache of network information is flushed), the conversations between the two machines happen across the single network. In your case if they happen to choose the wireless network, then you haven’t solved you problem.
Now, if the hardwired network is a different network … meaning that machines on the wireless network cannot connect to machines on the wired network, then adding both machines to the wired network may work, but only if you remove one of them from the wireless network, so they can’t “find each other” via the wireless path.
But my question is this: if the two networks are really the same network – meaning that you can get to all the same resources using either the wired or the wireless networks – and you need to connect to the wired network for speed – why stay connected to the wireless? Clearly you’re tethered by wire, so the machines certainly aren’t mobile.
I’d simply avoid the entire issue and disable the wireless adapters in the two machines, connect them both to the wired network, and be done with it.