A search for the production of bound tetraneutrons has been carried out with a time projection chamber using the reaction {sup 4}He({pi}{sup {minus}},{pi}{sup +}){sup 4}n at {ital T}{sub {pi}{sup {minus}}}=80 MeV and at 50{degree}{lt}{theta}{sub lab}{sup {pi}{sup +}}{lt}130{degree}. No evidence for tetraneutron formation was found, and a 90% confidence level upper limit for the production cross section of {ital d}{sigma}/{ital d}{Omega}{le}13 nb sr{sup {minus}1} was obtained.