Do you mean that, even though you're logging on with your saved password, you still get challenged for the password?
There is also a built-in auto-logout function but that triggers after no activity for some time (like, you leave your PC logged on while you have dinner or go out for a while) - I've not measured it, but it's at least an hour, possibly two.
How does the reducing attempt count show - its it something that's just displayed at login, like an announcement, that you have three chances at logging in or is it something you're seeing because your logins are failing?
Has that count been reducing over time - days, weeks?
I don't know if WP uses it, but a fairly common technique during many failed logins is to taper off the number of attempts allowed within some given time, or that further failed logins result in an increasing delay between later attempts.
But both of these depend on a series of failed logins, and that seems unlikely if you're using a stored password.
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Giraffe: a ruminant with a view.