OK, I've spent much of the afternoon trying to figure this out.

I think I've figured out what is going on.

Alaska Airlines Customer Service tells me that if you used a credit card for a certain number of transactions (she thought 3, but she wasn't sure), then the card is "blocked" until all those charges clear through the credit card company. For example, when you make a purchase, your credit card company shows it as "pending." If you have too many "pending" charges, then apparently you get the same message I've been getting:

"We are sorry but your credit card has been declined. Please select a different credit card and try again."

She said that you have to wait 24-48 hours for the pending charges to clear, then you can start all over again.

This is really a drag when you are trying to buy one-way tickets for several folks. That ends up being a lot of transactions!

Guess I'll just have to wait . . .