A series of polyimides of intrinsic microporosity (PIM-PIs) were prepared from commercially available rigid dianhydrides (pyromellitic dianhydride (PMDA), cyclo-[2.2.2]oct-7-ene-2-exo,3-exo,5-exo,6-exo-2,3:5,6-dianhydride (BTA), and 1,2,3,4-cyclobutanetetracarboxylic dianhydride (CBDA)) and diamines with norbornyl bis-benzocyclobutene (N2BC) segments (CANAL-2, CANAL-3, and CANAL-4). The physical and gas transport properties of these polymers were systematically investigated. The fractional free volumes (FFV), interchain distances, surface areas, and cumulative micropore volumes of these PIM-PIs were in the range of 0.156–0.185, 0.593–0.619 nm, 422–615 m2 g–1, and 0.059–0.072 cm3 g–1, respectively, showing a general trend of BTA > PMDA > CBDA for certain diamines. Thus, BTA-CANAL-2 concomitantly exhibited higher gas permeability and perm-selectivity relative to its analogues, viz., PMDA-CANAL-2, CBDA-CANAL-2, and 6FDA-CANAL-2 (6FDA: 4,4′-(hexafluoroisopropylidene)diphthalic anhydride). Specifically, it showed H2 permeability of 1685 barrer, H2/CH4 selectivity of 14, and H2/N2 selectivity of 15, which surpassed the 2008 Robeson’s upper bounds. The W-shaped, cage-like architecture of BTA residues imparted the resulting PIM-PIs with both high FFV and high cumulative micropore volumes, which was the reason for their excellent gas separation performance.