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But, to the real issue. I have probably owned at least 30 computers. NONE of them ever had more than one card. There really is no advantage. NIC cards seldom go bad. I don't even have two cards in my $4500 server. You could keep another card on hand. It will take you all of five minutes to install it. That gives you the advantage of an extra card for all of the computers rather than having to have an extra card in each.

Lol my past 3 have all had Dual NIC ports. They are useful depending on what you're trying to do. Sometimes you need to access 2 separate LANs or aggregate them to get better performance or use it for better performance. I often end up using LAG to increase the throughput of my file servers. Another common thing is to use them as failover as in this instance.

When you say switching from NIC to another. Are you unplugging the other NIC or are you just browsing using a different IP Address?

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the computer says "attempting to connect" but just does not connect to the SQL database

This part would be useful to see.