There are two systems to look at for this.
Too much blood loss
Each point of blood that's consumed deals one level of lethal damage to the victim. A strong, healthy vessel can take two or three levels of damage and be expected to recover just fine in time if they get proper rest.
But not every mortal victim has all seven of their health levels to begin with. Someone who is very small (children, dwarfs, petite adults) will have less blood to lose. Someone who is sick or injured will already have damage to their health track, meaning that drinking a point or two of their blood could push them over the edge. This is among the reasons why real-life blood banks have a questionnaire they ask before accepting a donor, and will reject some people.
Since vampires are rarely doctors and many medical problems don't have obvious external signs, an unlucky vampire can send a person to the hospital or kill them without intending to.
Page 285 to 286 of V20 covers mortal damage and healing rules.
Look at the chart for Frenzy on page 298. The smell, sight and taste of blood are all provocations that will call for a roll if the vampire is hungry. If a vampire hunts normally while they are especially hungry, they will be tasting blood once they bite into the victim. That can cause a Frenzy check. Once in hunger Frenzy, the vampire is going to drink quickly and greedily.
Ending a Frenzy is at the Storyteller's discretion, though one scene is typical. STs sometimes rule that a frenzy ends on its own if the provoking stimulus is removed. For example, you might be able to make a new Self-Control check to end a hunger frenzy once you have drunk enough to no longer be hungry. It is never guaranteed to work, and the ST may not allow it.
Even a round or two of hunger frenzy while feeding can be enough to kill a person, so ending a frenzy early won't always save the victim. A vampire can drink up to three points of blood per round (with a "normal" mouth. Certain deformities or shape-shifting can change this).
How hungry is hungry?
When a vampire's blood pool drop below (7 - Self-Control/Instinct) blood points, rounding up, they will be hungry enough to be making Frenzy checks. Getting near that amount the vampire feels uncomfortable and hungry, but not so strongly that frenzy is an immediate concern.
Because the Frenzy rules are explicitly left vague for ST interpretation, you should also talk to your ST about how they define "hungry" and when they'll generally be asking for rolls. If you're the ST, ask yourself these questions and consider how you want frenzy to impact your PCs and their story.
Extreme situations can and should bend the normal rules. The "sight of blood" will normally provoke frenzy only when hungry. However, if a vampire suddenly and unexpectedly finds herself standing in a room that's completely covered in fresh, glistening blood, that might be a strong enough provocation even if she isn't very hungry.