From an easily-available parts-breakdown illustration,* it seems that it's the thumb-stud on the magazine's follower, that actuates the pistol's hold-open latch.
So, my first question is: Does the magazine still have its follower's thumb stud?
Next, if the stud is there, does the bottom extension of the hold-open latch properly contact the follower's thumb stud? (Remove the left grip, and watch.)
Third, is the path of the pistol's slide clean and lightly lubricated?

If the hold-open works properly when you retract the slide by hand over an empty magazine, I would suspect either dirt or a too-strong recoil spring.
If the hold-open does not work properly when you retract the slide by hand, I would suspect either that the hold-open latch is broken, or that the magazine is.

In my personal experience, although not with anything like your particular pistol, I have found that difficulties like the one you describe are more often magazine-related, than they are ammunition-related.

*Fir the diagram, click on: http://stevespages.com/ipb-browning-buckmark.html
For the manual, click on: http://stevespages.com/page7b.htm, and look in the "B"s.
(No, I am not that Steve.)

Originally Posted by KevinC1966

. . . I thank you for any positive advise you may be able to offer.

I'm not stepping on Steve's usual good advice. Just another source of info.

The link below puts you directly into one of the very best Buck Mark sites. The guru is "Chim".

Ask your question (it's not an unknown problem). You will get good answers.
And most likely you will be directed to Chim's photo's on dissassembly.
Be careful taking off grips. Most do it inside a plastic bag. Springs fly.
And, the sticky "Buck Mark Starter Thread" at top contains a huge amount of Buck Mark info.

Browning - RimfireCentral.com Forums

Congratulations on having a Buck Mark. We belong to an exclusive cult, you know.

Yes, forum regulars are tired of this picture.
But, I can't resist showing my Fld Plus UDX with factory laminated rosewood grips.

Originally Posted by Steve M1911A1

From an easily-available parts-breakdown illustration,* it seems that it's the thumb-stud on the magazine's follower, that actuates the pistol's hold-open latch.
So, my first question is: Does the magazine still have its follower's thumb stud?
Next, if the stud is there, does the bottom extension of the hold-open latch properly contact the follower's thumb stud? (Remove the left grip, and watch.)
Third, is the path of the pistol's slide clean and lightly lubricated?

If the hold-open works properly when you retract the slide by hand over an empty magazine, I would suspect either dirt or a too-strong recoil spring.
If the hold-open does not work properly when you retract the slide by hand, I would suspect either that the hold-open latch is broken, or that the magazine is.

In my personal experience, although not with anything like your particular pistol, I have found that difficulties like the one you describe are more often magazine-related, than they are ammunition-related.

*Fir the diagram, click on: http://stevespages.com/ipb-browning-buckmark.html
For the manual, click on: page7b, and look in the "B"s.
(No, I am not that Steve.)

Thank you for the advice! I will try to work on the firearm later today and see if this works to remedy the problem. I really do appreciate your time and effort. I did print out the parts directory and manual. Again, thanks!

Originally Posted by DanP_from_AZ

I'm not stepping on Steve's usual good advice. Just another source of info...

Nobody who supplies useful information could ever possibly "step on my toes."

Originally Posted by DanP_from_AZ

Be careful taking off grips. Most do it inside a plastic bag. Springs fly...

Now, that's excellent advice!
I learned it the hard way, and spent a frantic hour on my knees on my basement's floor, finally finding an irreplaceable spring.
I should have included it, Dan, so I'm both mortified, and glad that you thought to.

In addition to what's been posted, take a close look at the slide lock, and the notch that accepts the slide lock. If the contacting surfaces are worn, the slide may not lock back. It's easy to wear alloy slides with steel parts if the slide lock is constantly used to release the slide instead fo "sling shotting" the slide to chamber a round. The thumb stud could be the issue, but I'd be surprised if it was. The stud on all my Buckmarks is a pretty stout piece of steel. Let us know what you find.

I didnt read all the responces, but it sounds like a magazine problem not a gun problem. tou may also try field stripping and pull the grips off and see if there is gunk in the slide lock.

Originally Posted by usmcj

. . . It's easy to wear alloy slides with steel parts if the slide lock is constantly used to release the slide instead fo "sling shotting" the slide to chamber a round. . .

I ALWAYS sling-shot the slide to chamber the first round.
I'll forgo describing all the experimenting done while trying to figure out "why" releasing the slide-lock would occasionally fail to feed.

During the 20 second timed-fire and 10 second rapid-fire phases,
I sure don't want a failure to feed/chamber on the "load" command.
I gave up "experimenting" with the slide lock feed reliability.
Doing the sling-shot works. Which is requirement number one. YMMV.