We study the effective field theory (including operators up to dimension five) of models in which dark matter is composite, consisting of either an electroweak doublet Dirac fermion (higgsino-like dark matter') or an electroweak triplet Majorana fermion (wino-like dark matter'). Some of the dimension-five operators in the former case cause mass splittings between the neutralino and chargino states, leading to a depleted rate of coannihilations and viable thermal relic dark matter with masses of the order of tens to hundreds of GeV rather than the usual pure higgsino thermal relic mass of 1 TeV. No such effects are found in the latter case (where the usual thermal relic mass is 3 TeV). Other operators, present for both wino- and higgsino-like dark matter, correspond to inelastic electromagnetic dipole moment interactions and annihilation through these can lead to viable models with dark matter masses up by an order of magnitude compared to the usual values.