One who is needy and refuses to accept help, it is as though he shed innocent blood (Jerusalem Talmud, Peah 8:8).
Maimonides extols what he calls the golden path, the middle way which a person should follow in life. He states that every trait has two opposite but equally undesirable extremes. The proper degree of any trait is not necessarily the median; it may be more toward one of the two poles, but it is never the extreme.
Self-sufficiency is certainly a desirable goal, and striving for independence is commendable. Some indolent people do not even try to carry their own weight. Their parasitism may be so reprehensible to other people that the latter may react by going to the opposite extreme and refusing to accept help when they need it. They may sustain physical injury by starvation or exposure, rather than accept a helping hand.
While accounts of great tzaddikim who subjected themselves to extreme degrees of deprivation do exist, these people had reached a level of spirituality so high that this deprivation would not harm them. For the average person, Solomon's caution, "Do not attempt to be too much of a tzaddik" (Ecclesiastes 7:16), should prevail. To do so may simply be an "ego trip." Some bridges can support vehicles of any tonnage; other bridges have a limit on the tonnage, lest they collapse under excess weight.
In this trait, like so many others, people may not be the best judge of their own capacities. Their best move is to seek competent spiritual guidance.