A gas turbine engine includes a compressor section, a combustor fluidly connected to the compressor section via a primary flowpath and a turbine section fluidly connected to the combustor via the primary flowpath. Also included is a cascading cooling system having a first inlet connected to a first compressor bleed, a second inlet connected to a second compressor bleed downstream of the first compressor bleed, and a third inlet connected to a third compressor bleed downstream of the second compressor bleed. The cascading cooling system includes at least one heat exchanger configured to incrementally generate cooling air for at least one of an aft compressor stage and a foremost turbine stage relative to fluid flow through the turbine section.