2007 Fit Sport A/T.
I don't drive this car normally, but have driven it.

Regular driver reports that twice, 3 months apart, the brake pedal got stuck while depressing, applying no braking power, then suddenly it would allow itself to depress the rest of the way and brake normally after making some popping or clicking noises from an unknown location. It has only been two freak occurances so far.

Here's what I noticed while I was unable to replicate the issue:

1. Brake fluid was clean and topped off above the max line.
2. With a helper repeatedly pressing the pedal with the car on, I see a few tiny bubbles rise up in the reservoir.
3. Where the master cylinder meets the brake booster looked as if it had been seeping.
4. Brake pedal used to be insanely responsive, now is a little spongey.
5. When in park, the shifter clicks when the brake pedal is pressed. (Normal?)
6. The booster, which I originally suspected when I heard the story, acted completely normal no matter how much I tried to rough it up with the pedal.
7. As I braked for the last time parking the car, I heard a couple not that abnormal popping/clicking type of noises from what seemed like the passenger front.
8. Uninstalled brakes, inspected, thoroughly cleaned, regreased, reinstalled. Found no issues.

I'm suspecting the master cylinder for the spongey pedal, but I'm lost for the actual reported issue.

 

Check your rear brake shoes and wheel cylinders,
Shoes may be out of adjustment
The wheel cylinders tend to seep some fluid under boots