The following discussion is controversial.
In less political terms, unbearable suffering is when someone decides their body/mind/life/world hurts too much, they are done with this life, and choose to die. That's pretty common with terminally ill people who are in agony. But it's not limited to that. Other common cases:
* Nonfatal, degenerative diseases that destroy the quality of life -- a leading example is dementia. Current technology cannot cure a lot of things. Some people don't want to spend a decade or more suffering before achieving natural release.
* Incurable mental or physical conditions that aren't lethal but are miserable -- a leading example is depression. People are urged to "get help" but some sufferers have been doing that for decades when they reach the conclusion that the medical industry cannot, in fact, help them and they wish to stop suffering.
* Having make mistakes so horrendous that they are not willing and able to live with them -- a leading example is causing the death of other people, especially loved ones.
* When a thorough examination of the options available to them does not reveal one they are willing and able to live with -- a leading example is suicide among doctors, lawyers, and other people whose education runs up debt so high they can only pay it by working that career, but then they discover it is soul-destroying.
* When it is not possible for them to obtains the means of a livable life -- a leading example is farmers in India drinking agrochemicals due to monstrous debt and inability to grow enough to sustain themselves and their family.
* When they find the world around them so abhorrent that they cannot in good conscience participate in it -- a leading example is the highest form of protest permitted to Buddhist monks.
There are, of course, many people who have suicidal thoughts but do not really wish to die. They deserve as much help as they want. In America at least, very few of them get help, which is a blight upon society. Of those who do, sometimes it actually works, they stop thinking about it, and feel very glad to have their lives back. That's great. But not everyone has problems that medicine or society can or will fix. They are then left to their own devices.
Most societies treat people as livestock in this regard. No amount of suffering justifies choosing to die. Society demands the right to torture people for decades, simply by refusing to let them leave. Religions often explicitly deny humans freedom of will over their own lifespan, assigning responsibility to some other entity considered more important. (Conversely, some other religions encourage suicide in certain situations, which is a different problem.) This creates a lot of tension, especially where different societies or religions disagree with each other.
Consent and body autonomy, however, paint a different picture. If you can't say no, then yes is meaningless. It is your body and you decide what happens to it. The most fundamental right of life is actually the right of death: the right to say, "I am done now. I am leaving." In fact, when people have that right, they are less likely to use it; they want the choice, so they can escape if the suffering ever becomes too much. Just that freedom makes suffering more bearable for many people. (Consider a comparison: people in charge of their own pain medication take less on average than when someone else controls that.) Choice matters that much to people.
If society does not respect people's freedom to die, then some people will take it by force. This denies them the right to a reasonably painless death with dignity, having made what final arrangements they can, and having enjoyed a chance to say goodbye if they have friends or relatives. People who have benefited from assisted dying have expressed gratitude for these things, and people forced to flee their bodies in secrecy have condemned being cheated of them -- literally, treated as less than animals, which are granted a peaceful death if they cannot be healed.
And so, like sex, it's not a matter of whether this happens. It always happens. Some people will choose to die rather than suffer, regardless of the pressure applied to force them to remain. What we have left to decide is how comfortable and humane that passage will be.