The recoil distance method has been used to measure the mean lifetime of the J/sup ..pi../ = (9/2)/sup +/ 1755-keV level of /sup 21/F, formed (predominantly) via the /sup 7/Li(/sup 18/O, ..cap alpha..)/sup 21/F fusion-evaporation reaction. The lifetimes of the 984- and 1309-keV levels of /sup 20/F and the 1275-keV level of /sup 22/Ne were obtained coincidentally. Similar measurements in the /sup 7/Li(/sup 11/B, pn)/sup 16/N reaction determine the lifetimes of the 298- and 397-keV levels of /sup 16/N. Data were acquired at theta/sub ..gamma../ = 0/sup 0/ using an escape suppression spectrometer, with /sup 18/O and /sup 11/B incident energies of 28.5 and 15.3 MeV, respectively. The experimental results are the following: /sup 16/N(298,3/sup -/), tau = 133 +- 4 ps; /sup 16/N(397,1/sup -/), tau = 6.60 +- 0.48 ps; /sup 20/F(984,1/sup -/), tau = 1.8 +- 0.3 ps; /sup 20/F(1309,2/sup -/), tau = 1.9 +- 0.3 ps; /sup 21/F(1755,(9/2)/sup +/), tau = 3.5 +- 0.5 ps; /sup 22/Ne(1275,2/sup +/), tau = 5.1 +- 0.2 ps. Shell-model calculations for /sup 16/N and /sup 20,21/F have been performed and are compared to experimental results.