We present deep observations of the nearby T-type brown dwarf binary $\epsilon$ Indi Bab in radio with the Australia Telescope Compact Array and in X-rays with the Chandra X-ray Observatory. Despite long integration times, the binary (composed of T1 and T6 dwarfs) was not detected in either wavelength regime. We reached $3\sigma$ upper limits of $1.23 \times 10^{12}$ and $1.74 \times 10^{12}$ erg/s/Hz for the radio luminosity at 4.8 GHz and 8.64 GHz, respectively/ in the X-rays, the upper limit in the $0.1-10$ keV band was $3.16 \times 10^{23}$ erg/s. We discuss the above results in the framework of magnetic activity in ultracool, low-mass dwarfs.