We present H I21-cm line observations, obtained with the Australia Telescope Compact Array, of two dwarf elliptical galaxies (dEs) in the Fomax cluster: FCC032 and FCC336. The optical positions and velocities of these galaxies place them well within the Fornax cluster. FCC032 was detected at the 3σ significance level with a total HI flux density of 0.66 ± 0.22 Jy km s -1 or an HI mass of 5.0 ± 1.7 x 10 7 h -2 75 M ○. . Based on our deep Ha + [N II] narrow-band images, obtained with FORS2 mounted on the Very Large Telescope, this dE was already known to contain 600 -1800 h -2 75 M ○. of ionized hydrogen (depending on the relative strengths of the Hα and [N II] emission lines). Hence, this is the first study of the complex, multiphase interstellar medium of a dE outside the Local Group. FCC336 was detected at the same significance level: 0.37 ± 0.10 Jy km s -1 or a total H I mass of 2.8 ± 0.7 x 10 7 h -2 75 M ○. . Using a compilation of H I data of dwarf galaxies, we find that the observed high HI mass boundary of the distribution of dwarf irregulars, blue compact dwarfs and dwarf ellipticals in a log L B versus log M HI diagram is in good agreement with a simple chemical evolution model with continuous star formation. The existence of many gas-poor dEs (undetected at 21 cm) suggests that the environment (or, more particularly, a galaxy's orbit within a cluster) also plays a crucial role in determining the amount of gas in present-day dEs; for example, FCC032 and FCC336 are located in the sparsely populated outskirts of the Fomax cluster. This is in agreement with H I surveys of dEs in the Virgo cluster, and an Ha survey of the Fornax cluster, which also tend to place gas-rich dwarf galaxies in the cluster periphery.