There is a close relationship in high temperature superconductors between the carrier density, the magnetic response, and the unusual d-wave pairing. One of the most surprising results in the underdoped region of the phase diagram is the occurrence of a marked decrease in the charge scattering rate below a frequency scale associated with a pseudogap in the electron spectrum. The origin of the pseudogap and its doping dependence are subject of continuing controversy.  I will present neutron scattering results in the bilayer YBa₂Cu₃O_6+x system with a maximum superconducting transition temperature of 93 K. Particular attention will be paid to the regime with a low density of holes (x = 0.35 and 0.5).  I will make a comparison to single-layer cuprates and discuss possible origins of the pseudogap.