Neti neti, "not this, not this", is the method of approaching the ultimate by negation, and it is among the Upanishads' most distinctive contributions. Its source is the Brihadaranyaka Upanishad, where Yajnavalkya, pressed to describe the self, answers that it can be described only as "not this, not this", because there is nothing beyond this negation (2.3.6, and again at 3.9.26 and 4.5.15).
The underlying logic is that whatever can be pointed to as an object is, by that very fact, not the subject doing the pointing. The self is the witness, and a witness cannot be found among the things witnessed. Every candidate is therefore set aside, body, breath, mind, intellect, and every state, including deep sleep, not because these are unreal but because none of them is what the question was about. The Taittiriya Upanishad's analysis of the five sheaths performs the same operation systematically, moving inward through food, breath, mind, understanding and bliss, and the Mandukya's treatment of waking, dream and dreamless sleep does it again in another register.
Advaita Vedanta made this its principal method of instruction, and Shankara uses it constantly. It is important that the tradition pairs it with the positive formulations rather than opposing them: negation clears away misidentification so that statements such as "you are that" and sat-chit-ananda can be received without being converted into descriptions of an object. A method of pure negation would be as much a failure as a naive positive description. Ramana Maharshi's twentieth-century teaching of self-enquiry, the question "who am I?", pursued by setting aside every answer that names something observable, is a direct continuation, and is largely responsible for the method's currency in the modern West.
Analogues elsewhere are real but should not be over-assimilated. Christian and Jewish apophatic theology, in Pseudo-Dionysius and in Maimonides, proceeds by denying attributes of God; Islamic tanzih does something similar; and Buddhist Madhyamaka's fourfold negation refuses all four possible positions on a question. The family resemblance is genuine, but the aims differ: apophatic theology denies attributes of an object of worship, neti neti denies objecthood to the subject, and Madhyamaka denies inherent existence to everything, including its own negations. The differences matter at least as much as the similarity.