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At which point does being out of tune become a problem?

You have to be in tune with the other instruments. Even in synthesis, the mathematical relationship of the tuning determines the sound. Even if you want a discordant sound, the amount out of tune one wave is from the other will shape the nature of the sound.
 
This is how the frets would look on a guitar that can be tuned more precisely, (true temperment) than one with the customary fret position/intonation compromises found on most guitars.
View attachment 7138640
Sorry if this post does not answer the original question.
True temperament for whay key?

BBB
 
If you are in tune with yourself, and playing alone, then it's not a problem. When playing with others, it's a problem, especially if one of them has a digital piano. Also, imagine you are tuned a little flat and the guitarist is tuned a little sharp... the difference might be noticeable and unpleasant. I had an upright bass teacher who said, "resist the temptation just to pick it up and play without checking the tuning."
 
I feel like I can be out by something like 20 cent and it's still just fine.

What are your experiences with this?

As someone with very high sensitivity to intonation, ±20 cents all the time would drive me up the wall. Anything more than ±5 and my ears start to twitch. With that said, there is a big difference between 'sweet intonation' and 'being out of tune', especially when lots of continuous-pitch instruments (or voices) are playing together.
 
“When does it become a problem?”

Immediately.

Folks who’ve never played with an orchestra or even a larger band might not realize how much “power” is lost to a band’s sound when there are tuning issues, but play that concert Bb when everyone is in tune and you feel it in the floors and the walls of the auditorium.

Playing bass on your own might not be as immediately apparent if you don’t have good ears, but even playing chords on a solo guitar that’s slightly out of tune vs one that is in tune will also demonstrate the power lost from tuning problems. So yeah, it’s an immediate problem and the reason why most teachers will say the first thing you need to do is tune, warm up, tune again, and then begin actual practicing.
 
As someone with very high sensitivity to intonation, ±20 cents all the time would drive me up the wall. Anything more than ±5 and my ears start to twitch. With that said, there is a big difference between 'sweet intonation' and 'being out of tune', especially when lots of continuous-pitch instruments (or voices) are playing together.
this is me. even if things are off my a tiny increment it drives me nuts. When famous songs are played in different keys I go berserk too. It's an affliction
 
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I have tried to test my hearing by tuning my bass open A string with a clip-on tuner, and then tuning the other strings by ear, and then checking the result of my tuning by ear with what the clip-on says. When I do this, I find that I am (usually) pretty close to what the Snark says.

Without creating ones own way to test the ability to discern when, (for the sake of simplicity) two simple sine wave tones, (not complex sounds with overtone content like a musical instrument produces) are either the same or not by varying degrees.

Does anyone know of some sort of app designed to test just how far "out of tune" a sound must get for me to say I am able to recognize it is so? I envision something like two tones are produced, which could also be octaves apart, and you then say if they are, or are not in tune.
 
...

Does anyone know of some sort of app designed to test just how far "out of tune" a sound must get for me to say I am able to recognize it is so? I envision something like two tones are produced, which could also be octaves apart, and you then say if they are, or are not in tune.

Just for a laugh, have a go at this:
 
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I am really sensitive to “out of tune”. Makes it hard to play along with certain recordings. Even a chorus pedal can bother me. Playing live I sometimes have to say “f it” and get over guitar being out and just play along bending notes to relative pitch until it’s over. No one in audience cares anyway. Lol
 
It depends on the musician's ear in question who is interpreting the tuning, their ability to discern how 'out' something is, and the reference to which everyone is tuned. You can't use a tuner perfectly set at A440 and then play with a piano that hasn't been tuned in a while where 'A' is 432 and expect to sound great - you have to tune to the piano in that instance. Some people don't tune to 'A440' out of choice, and our modern systems of tuning are not 'perfect', and neither are our instruments - especially those which aren't fixed-pitch. I have unfortunately played with people before who really can't tell they are out of tune and are not making any effort to address it, and it is painful.

As someone who plays upright AND fretless, I am very 'in tune' (pun intended) to relative pitch across different instruments and tuning in real-time. The longer you play, the more sharpened this skill becomes, and the less tolerant you are of yourself and others being out of tune. While it may not be as apparent at times in an ensemble mix (you'll definitely 'feel' like something is off somewhere but you can't pinpoint it), if you are playing unison lines with another instrument or doubling/harmonizing melodies, you won't be able to ignore it.

My sax player and I have been doing a bunch of duo recordings lately and the days where we are both really in tune and on point together are magical. The days where one of us is struggling, or if the weather is causing tuning issues, can be a bear, especially for a sax player. Even being a very small percentage out of tune can be very noticeable and cringe-inducing. When you spend a lot of time in the studio and in post-production doing mixing and mastering, tuning discrepancies become very apparent.

None of us are perfect, but we should all be practicing with a tuner or other fixed pitch reference and refining our intonation, and developing our ear to the point where we can improvise/correct it in the moment as much as we can when things go sideways.
 
This is how the frets would look on a guitar that can be tuned more precisely, (true temperment)
Posts mentioning true temperament/just intonation often omit the critical data point:
true temperament only works in one specific key, any other keys will sound worse

even/equal
temperament is the standard because is equally, slightly , out of tune in all keys.
 
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Yes, what no one wants to mention when they repeat that old line about "F# and Gb are different" is that it TOTALLY depends on context. Yes, in "just" temperament the F# that's the Pythagorean third of D major will have a slightly different frequency than the Gb that's the 11th of Db major. But the third of Bb (D) third of A# (C##) are the same thing.

I disagree with the characterization of equal temperament as everything being slightly out of tune. Yes, it's true that fifths, fourths, and other intervals aren't integer multiples of each other; but that's not the only possible definition of being "in tune". In-tuneness can also consist of being integer multiples of the twelfth root of two apart.