The weakly interacting BNg and BNg(2) molecular systems, Ng = He, Ne, Ar, Kr, and Xe, have been thoroughly studied through coupled-cluster RCCSD(T) calculations and large correlation consistent basis sets. For the BNg diatomics, the states examined are the X-2 Pi and A(2)Sigma(+), and the (X) over tilde B-2(1) state for the C-2v BNg(2) triatomics. A series of corrections render our final results reliable, judging as well from the (limited) experimental numbers available. Both BHe and BHe2 are marginally unbound, whereas the attractive interactions of the BNg X-2 Pi states, where Ng = Ne, Ar, Kr, and Xe, are D-0 = 19.8, 98.2, 141.9, and 209.1 cm(-1), respectively. For the BRn (Rn = radon) species, an estimated value of interaction energy D-0 approximate to 280 cm(-1) is obtained by a D-0 versus static polarizability (alpha) extrapolation. Corresponding atomization energies of the BNg(2) ((X) over tilde B-2(1)) molecules are AE(0) = 52.0 (BNe2), 263.4 (BAr2), 384.6 (BKr2), and 576.9 (BXe2) cm(-1).