If someone is feeling bad at high altitude, how do you tell if their condition is so severe that it threatens their life unless they descend immediately?
Many people at moderate altitudes (about 10,000-13,000', 3000-4000 m) experience mild AMS (acute mountain sickness). The most common symptom is a headache. Mild AMS is not life-threatening, and people may experience similar symptoms due to other factors that are common in a mountain environment, such as lack of sleep, caffeine withdrawal, sunburn, or unaccustomed exertion. One study (Dallimore) of people who hiked at elevations below 1500' (500 m) showed that about 7-11% of them had "symptoms that at high altitude would lead to a diagnosis of AMS," defined by scores of 3 or higher on the Lake Louise questionnaire. Because of the high rate of false positives and the possibility of lumping together disparate conditions and symptoms, questions have been raised about whether the Lake Louise questionnaire is a good method for diagnosing AMS (ISMM, Hall, MacInnis).
Two non-symptoms are fainting and difficulty sleeping. Although poor sleep has in the past been used as a factor in diagnosing AMS, recent work (MacInnis) shows that taking it into account makes for a less accurate diagnosis. Of course, poor sleep will cause you to feel lousy, and mild AMS is basically a condition in which you feel lousy. Sometimes people will faint at high altitude when they try to stand up suddenly; this is fairly common, and is not a sign of anything serious if there are no other symptoms.
Severe altitude illness is another matter. There are three common forms of severe altitude illness. All of these can be life-threatening and require immediate descent. They occur most commonly above about 15,000' (4600 m).
The most common sign that distinguishes severe AMS from a mild case is ataxia, which means general problems with coordination and balance. A good test for ataxia is to have the person walk heel-to-toe in a straight line, as in a drunk-driving test. Ataxia is a sign that something is wrong with the person's brain. Other signs of an altered mental state (irrationality, changed behavior, lethargy, babbling) may also be signs of severe AMS.
In addition to these mental symptoms, people with severe AMS will often have some of the other symptoms of AMS in such severe form that they become incapacitating. These include headache, nausea and vomiting, weakness, and dizziness. If any one of these symptoms becomes incapacitating, then severe AMS is likely.
High-altitude cerebral edema (HACE)
HACE is a condition in which fluid leaks into the brain, causing it to swell and become compressed within the skull. It is similar to severe AMS and probably overlaps with it. The effects can progress until the person becomes incapacitated or goes into a coma.
High-altitude pulmonary edema (HAPE)
This is a condition in which fluid leaks into the lungs and keeps the person from breathing normally. They are unable to catch their breath, even when resting, and can't complete a sentence because they run out of breath. They may have rales, meaning a crackling sound when they breathe. To check for this symptom, have the person take several deep breaths first, and then press your ear tightly underneath their right arm at nipple level and listen as they keep breathing. Rales do not indicate HAPE if they occur without other symptoms or during the first few deep breaths.
Bezruchka, Altitude illness: prevention and treatment, 2005
Dallimore, Foley, and Valentine, "Background rates of acute mountain sickness-like symptoms at low altitude in adolescents using Lake Louise score," Wilderness Environ Med. 2012 Mar;23(1), http://www.ncbi.nlm.nih.gov/pubmed/22441083
ISMM World Congress in Bolzano online discussion session, 2014, http://www.altitude.org/ams.php
Hall et al., "Network Analysis Reveals Distinct Clinical Syndromes Underlying Acute Mountain Sickness," PLOS One January 2014, http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0081229
MacInnis et al., "Is Poor Sleep Quality at High Altitude Separate from Acute Mountain Sickness? Factor Structure and Internal Consistency of the Lake Louise Score Questionnaire," High Altitude Medicine and Biology, December 2013, 14(4): 334
As a general rule: Assume acute mountain sickness (AMS) unless proven otherwise. AMS is potentially life threatening, so if you have symptoms related to AMS, do not ascend any further. If they do not get better, descend.
Low altitude is not an indication against AMS which can occur above 2000m already. So the assessment should rely on your symptoms. Early symptoms of AMS are 
- Difficulty sleeping
- Dizziness or light-headedness
- Loss of appetite
- Nausea or vomiting
- Rapid pulse (heart rate)
- Shortness of breath with exertion
Obviously, the last two occur anyway at high altitudes, so you have to judge whether it is pathological. To help asses how bad the condition is one can use the Lake Louise Score. This is not a definitive diagnosis, but a way to objectively judge your condition. This is fairly conservative, so if you follow this advice you should never get into a situation where you develop more advanced symptoms, High-altitude cerebral edema (HACE) or High-altitude pulmonary edema (HAPE). The latter two are actually symptoms of AMS which are directly live threating and therefore often referenced as conditions of their own. Advanced symptoms include: 
- Blue color to the skin (cyanosis)
- Chest tightness or congestion
- Coughing up blood
- Decreased consciousness or withdrawal from social interaction
- Gray or pale complexion
- Cannot walk in a straight line, or walk at all
- Shortness of breath at rest
If you experience these, the condition is already life threatening and you need to descend asap.
The usual case is, that you have a mild form of AMS not including any advanced symptoms and you need to make a decision on how to proceed. This depends a lot on logistical factors: Where are you, where do you want to go and how easy can you descend. If your plans allow it, the best option is always to stay and see how the symptoms progress. If you need to go on, you have to factor in escape plans. Assuming you can easily descend, going on is save, just keep track of the symptoms and react accordingly. If going on means committing for a push into an area where the only way out is walking, the save way to go is aborting. The symptoms can get worse fast and impact your ability to walk out severely. This means you will stay longer at high altitude which again makes the AMS worse.
 J. Heller, Acute Mountain Sickness, 18.10.2015, https://www.nlm.nih.gov/medlineplus/ency/article/000133.htm.
Mainly used for the symptom lists, as pointed out in the comments some parts of the article are not very realistic.
SGGM, Akute Höhenkrankheit, syn. AMS, (Acute Mountain Sickness), 18.10.2015, http://www.mountainmedicine.ch/index.php/de/medizinisches/ams.