Encroachment

During Ichigo Kurosaki's training with Kisuke Urahara, he went through a particularly lethal lesson called Shattered Shaft. He had to escape the shaft in time to gain his own Shinigami powers, or else he would become a Hollow.

Pluses
The Encroachment is when a plus begins the process of transforming into a Hollow. Once the Chain of Fate is separated from a spirit, they can no longer return to their physical body. Death for the soul comes once the chain is completely eroded. One symptom right before the last erosion is an increase in appetite, as stated by Jinta Hanakari. The last erosion is much more severe than the others, as the final erosion experienced by Ichigo involved seven links of his chain, as opposed to the one link per erosion before it. They become Hollows afterwards, and have no chance of returning to their spirit form - the transformation is permanent, unless the soul is purified by a Shinigami's Zanpakutō so it can go on to Soul Society. When a Plus descends into a Hollow, his/her spirit form shatters and reconstitutes into a different form. Then their Hollow Mask manifests.

Resistance
Urahara replied when Ichigo asked if he was trying to kill him, "Yes....the moment you give up," implying the Encroachment can only be overcome by those who have the will to do so. While the standard plus usually becomes a Hollow straight away, Ichigo Kurosaki was able to fight it if only long enough to defeat it. The transformation went completely out of sequence: His mask was being born first while his body was still a plus. The white substance forming the mask was almost pouring out of his eyes and mouth and began to cover the right side of his face. Had the process continued it would have completely formed with no chance of returning to a plus. This ability to resist may or may not be implied for all souls, just those similar to Ichigo Kurosaki (see below.)

People with Latent Shinigami Abilities/High Spiritual Power
The real test was not to escape the shaft, but for Ichigo to become a Shinigami while simultaneously preventing his own Hollowification - if a soul were to reach the encroachment and gain Shinigami powers in time, they can prevent their death. Ichigo found the strength to resist the transformation and find his own Zanpakutō, Zangetsu, just in time.

Ichigo, however may be an exception as he was born from a Human mother (Masaki Kurosaki) and a Shinigami father (Isshin Kurosaki), so he was born with the potential to become a Shinigami, though his true heritage was not revealed until the Arrancar Arc.