I get that you should bid what you can afford, but sometimes that changes as cycles go by. The following has happened to me more than once:
Earth Day 1 - 7am - International Prospect Discovery announced
Earth Day 1 - 11am - I bid on IFA. I'm not in first. Bummer
Earth Day 1 - 3pm - I raise my bid by $1M. Huzzah! I'm in first.
Earth Day 2 - 3pm - I'm still in first, same bid
Earth Day 3 - 3pm - I'm still in first, same bid.
Earth Day 3 - 7pm - I suddenly lose the prospect off the board to a bidder on that cycle.

Which one of the following is happening:
A) There is a way to tell what the last cycle is on an IFA, and certain owners know it.
B) Raising a bid by a certain percentage over the winning bid after a certain period of time ends the bidding.
C) Other? Not sure what I'm doing wrong, but it's annoying when you don't have the opportunity to counter.

I long suspected that each IFA has a predetermined signing time. (No such thing as "bidding wars"; no amount that triggers signing.) Awhile ago I was also told that a developer chat once confirmed this, for what that's worth.
If you really want a player, bid what you think will win.
If you don't mind potentially losing him, bid what you think he's worth.
If you want to see if you can get him on the cheap, take your chances.
I've done all three.

Thx -- so the question becomes, is there a way to get a hint as to when the bidding will end?

I don't believe so. Never heard of any owner saying anything like it. To use your first example, you lost out to another owner without any chance to counter. He could've been bidding on the player all along and just outbid you at the signing time. He could've only just decided to go after that player, maybe even flipped money from another player to do it. It's even possible the system didn't show him the player till long after you saw him.

I think your last posit is the most possible, as he outbid me by around 50%, which made me believe perhaps there's a threshold with a late bid over a certain amount above the next for the bidding to end.

If there is a way to tell how long a player will remain available, I doubt it will get posted here.

I know I was on the other side of it recently. I was bidding on a player and wasn't willing to go any hire on him so I withdrew my bid on him and put it all on another player and I got him the next cycle. That one I think was pure luck and coincidence that I just bid on him for the first time on his signing cycle.

I think some players want to wait longer than others and there's no way to tell. I've had the highest bid on players from beginning to end where I've gotten them. And sometimes that beginning to end is 2 days and sometimes almost a week

Don't forget what happened a few years ago. In case you missed it, I'll recap:

One of our owners figured out that, once he saw his first IFA at the beginning of a season, he could replace the numbers at the end of the player card's URL, and by extending the sequence, see all the IFAs - in the order that they'd enter the IFA pool - that were due to come out that season. Only thing he couldn't see is the exact dates they'd come out.

Don't bother trying it now, the exploit was blocked as soon as it became public knowledge. We complain about how long it takes support to do things around here; let me tell you that was the fastest I've seen them plug a leak of any kind.

So think about it - if the entire IFA pool is generated from the very beginning of the season, and their sequence of entry into the pool is pre-programmed, doesn't it make sense that the simplest way to program their entry and signing dates would also be to predetermine them?