The question, as stated is irrational. One has the right to die, because one cannot have it taken away. If you want to die, and can act to kill yourself, there are a legion of ways to accomplish it. So, it’s not that suicide is difficult to accomplish nor that people fail to. Instead, it is difficult to understand how those that say that they wish to, can somehow fail to. And therefore we argue that if you wanted to you could and would. So that like many human utterances, a desire for suicide may be a situational expression of emotional frustration, pain and exhaustion, rather than a desire to end one’s life. It is impossible for others to know. And like many thoughts of human beings, it may be impossible for the individual himself to know.
That said, the moral and legal prohibition in the west, is against assisting others with suicide, not against suicide itself. The moral counsel against suicide exists only to encourage those who consider suicide to seek assistance in removing the source of their suffering rather than resorting to suicide.
So instead of asking why one may or may not have the right to die, the question must be “Why can others not kill me if I desire it?” And the reason is that it is an irreversible decision, and we cannot ever really know whether it is a true desire unless one performs the act one’s self. It may be simply that the individual is venting exhaustion or frustration at that moment but at somepoint would change his mind. And it is unjust to ask others to bear the risk of participating in a decision that we cannot know. Because the only demonstration of a human beings preferences is not his verbal statements but his actions.
Secondly, as someone says above: “an elderly terminally ill patient may feel an obligation to commit suicide to relieve their family and the society of them as a perceived burden.” Which obscures the more common and problematic circumstance: that the elderly patient who is terminally ill, or even, or who is a substantial burden on family and society, will be pressured to commit suicide and comply despite their desire not to, and to hold onto life a little longer.
Therefore the moral and legal prohibitions against assisting with suicide are rational in philosophical and practical terms.
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Afterward: It is somewhat interesting to see among these answers, the prevalence of the misapplication of scientific principles to a problem of moral exchange. The inability to tell the difference between a problem that has can be objectively resolved by relying on an analysis of outcomes, rather than one consisting of an exchange between individuals that can be intersubjectively tested. That our citizens no longer rely on scriptural analogy may be beneficial. But that they do not see morality as a system of exchanges, rather than a means of achieving a utilitarian end, is somewhat more frightening than the misapplication of those moral principles to questions about the physical world.