I do this with IFAs more than with regular FAs. With IFAs, if it's a great player, I assume most teams that can see him will be at their max bid amount within 5-6 cycles. So if I'm ahead after two days, it's not likely someone is going to be able to jump in and outbid me (unless they were on vacation or something). So I'll scale back a million or two to try to save money. If I'm suddenly not first place anymore, no problem. I'll just go back up to my original offer.
With veteran free agents, everyone can see them, including all the teams with more money than me. So if I'm in first place, I'll usually stay at my current offer. Since there are so many teams and so many free agents available at once it's always possible that another team that was only willing to offer $10m per year will lose out on another of their targets and suddenly be willing to offer $12 million on day 3.
And I have done the same thing as Mike. If you don't have a backup option in mind, you might as well leave your max bid on the table in case the higher bidder drops out.