Pneumatic opens with pn-, a cluster English will not pronounce, so the p goes silent. That is a Greek signature, matching ps- in psalm, pt- in pterodactyl and mn- in mnemonic. Greek was comfortable with these openings; Latin transliterated them faithfully; English inherited the letters and quietly dropped the first sound.
The word comes from Greek pneumatikos, from pneuma, "wind" and also "breath." That double sense matters, because in Greek the word is attested mainly in its other meaning: of spirit, spiritual. Breath, wind and spirit are one word in Greek, as they are in Latin spiritus and in Hebrew ruach. English took pneumatic in the 1650s in the strictly physical sense of belonging to air or gases, leaving the spiritual half of the Greek word behind.
pneumatic (adj.) "moved or played by means of air; of or pertaining to air or gases," 1650s, from Latin pneumaticus "of the wind, belonging to the air," from Greek pneumatikos "of wind or air" (which is attested mainly as "of spirit, spiritual"), from pneuma (genitive pneumatos ) "the wind," also "breath" (see pneuma ). Earlier was pneumatical (c. 1600). The pneumatic-dispatch tube was so called by 1859 (in Paris, pneumatique ). also from 1650s