Question author says the following:
I'm not in academia, and I confess I only consider this function in the context where it's News due to fails - ie, "So & So breakthrough is published, but other labs were not able to reproduce...". Nor do I recall a popular science headline about a discovery, that bothers mentioning successfully reproduced results.
In addition to the other answers, I'll try to provide some (potentially) instructive/informative examples.
Reproducibility is the hallmark and cornerstone of science!
What separates science from other forms of explanations of the world around us and for example the healing arts is reproducibility. If I explain to you how to do something in detail, all my techniques and precautions, and what results I get along with my estimated errors, then you should be able to follow my instructions and get the same result, within some range of variation that is consistent with recognized estimates of experimental error.
If I say that mixing blueberry juice and thumbtacks makes plutonium and tell you how I did it, then you should be able to try to make plutonium too. If it doesn't work for you, my results are irreproducible, and there's a journal for that.
Here are a few extreme examples that may bracket how the reproducibility (or lack thereof) of a given result was or was not confirmed in a timely way.
Alzheimers - Tragically, NOT enough effort was spent trying to verify a result, time & lives wasted?
Biology SE's Does the recent concern over several papers about Aβ*56 call into question the association of Alzheimers Disease with any amyloyd beta oligomer forms? gives more details on a really tragic failure of science to police itself.
Not enough effort was spent trying to reproduce an experiment pointing to a certain form of amyloid protein's link to Alzheimer's disease. Instead many biologists spent years doing not-so-fruitful research based on the premise that this result was correct and several pharmaceutical companies spent (essentially our1) money and (their) time developing drugs and doing drug study after drug study that did not demonstrate results.
The scientific community failed us in this case because they did not confirm that the original results were reproducible.
Several papers were retracted years later, some by the journal without the authors' input. There was plenty of damage to go around, but no way to recover the time and money misspent.
GSI positron peaks
The positron-electron peak puzzle: results from APEX (Ahmed et al (1997) Z. Phys. A 358, 235–236 (1997)) summarizes a few million dollar project at Argonne National Laboratory to try to verify some strange results seen for years by two competing groups at GSI Puzzling Positron Peaks Appear in Heavy‐Ion Collisions at GSI (Physics Today v. 38, no. 11, p. 17 (1985)) There was no room for such peaks in modern understanding of physics, they stubbornly remained at the few standard deviation level and release of specific details of analysis (and the raw) data was never completely forthcoming.
It was irritating to nuclear and particle physicists, so a clean start on a different continent by a group with no ties to GSI was initiated. A huge amount of effort went into trying to reproduce the results, including consultations with those groups to make sure nothing was omitted.
Cold fusion
This is another "irritating" result for which there was no room in Physics for the phenomenon, but the potential payoff was incredible.
Unlike the GSI positron peaks, this one was relatively easy to approximately reproduce - you could stick some palladium in water, run a current through it and put a neutron detector next to it or put it in a calorimeter and measure the heat produced compared to electrical power, but in this case due to greed the originators withheld certain specifics that made honest attempts at reproducing the results unable to implement tests that were exact reproduction.
Lack of sufficient information to test the reproducibility fed the storm and the pockets of some individuals.
1Research budgets of established pharmaceutical companies comes from revenue - i.e. how much we pay directly or indirectly for the medicines that keep us alive and healthy. This also means time and money was NOT spent on potentially more helpful drug development.