Is it possible to condition oneself to not be bothered by distressing sounds? It is my (layman's) understanding that a domestic cat learns to meow in a specific tone (I've even read that it can resemble a human baby crying) in order to get the owner's attention.

Regardless of the factual accuracy of this specific claim (which is referenced here, here and was even challenged in Skeptics.se), the fact is that an attention-seeking cat's meowning can be really unsettling.

Again, speaking as a layman - and to give a practical example -, when I'm trying to concentrate on something and my cat meows two or three times (even at a great distance), it is more than enough to disrupt my attention completely.

(More context - can be skipped)


  My strategy to deal with this has been mixed: Sometimes I'll ignore
  him for a moment - which is incredibly hard; sometimes (if it won't
  stop) I'll yell (not too loud, but very firmly): STOP. And it works.
  
  As of lately I've been interested in dealing with the root cause.
  Precautions have been taken to assure that the cat is getting enough
  food, water, grooming, and indeed, enough attention.* I've been
  struggling to not scream (because a) it makes me feel bad, and b) it
  sure reinforces the cat's behavior).
  
  (*) While this could be debated, because one would think that
  the cat wouldn't meow if he didn't need to, I'm quite sure he's
  getting at least the most attention I can provide. From my research it
  should be more than enough. And it is enough for my _other_ cat, as a
  matter of fact.


(End "more context")

To be pragmatic, while the cat's behavior cannot be effectively resolved, it is of my utmost interest to learn if the owner can "recondition" him/herself not to be bothered by the sound of the meowning so much.

I'm wondering if a person can be made to simply "observe" the distressing sound, like a Zen Master observes the sound of the wind rustling the leaves of a tree, without her brain generating an immediate distress signal like it does.

Or, maybe, some tones are just "hardwired" to always trigger such reaction, no matter how much cognitive effort a person puts in?
Short answer
Meditation may be key to block out distracting noise. 

Background
One study has shown that during meditation, expert meditators with over 19,000 h of meditation experience show less brain activations in regions associated with discursive thoughts and emotions (prefrontal regions, basal ganglia, and subthalamic nuclei), and more activation in regions related to response inhibition and attention (ventral prefrontal cortex and intraparietal lobule). 

The authors used neutral (restaurant background noise), positive (cooing baby) and negative sounds (a screaming lady).

However, meditation is probably not the conditioning training you are after for use in everyday life, but since the question has been silent I thought to start off with this lead.

Reference
- Brefczynski-Lewis et al., PNAS (2007); 104(27): 11483-8