We propose that the transition temperature in the layered cuprates is affected by a peculiar interplay between doping and pair-breaking scattering. This scattering can lead to a depression of Tc relative to its ``intrinsic'' value. Using this picture, one can explain the large value of Tc in the three-layered cuprates relative to the two- and one-layer cuprates and also understand the large pressure effects observed. We predict that one can raise Tc above the usual maximum value by applying pressure on underdoped TI-2201 samples and the maximum value of Tc under pressure is evaluated. The temperature dependence of the pressure coefficient is also discussed.