An apparatus and method are provided for processing instructions fetched from memory. Decode circuitry is used to decode the fetched instructions in order to produce decoded instructions, and downstream circuitry then processes the decoded instructions in order to perform the operations specified by those decoded instructions. Dispatch circuitry is arranged to dispatch to the downstream circuitry up to N decoded instructions per dispatch cycle, and is arranged to determine, based on a given candidate sequence of decoded instructions being considered for dispatch in a given dispatch cycle, whether at least one resource conflict within the downstream circuitry would occur in the event that the given candidate sequence of decoded instructions is dispatched in the given dispatch cycle. The dispatch circuitry has resource checking circuitry arranged, by default, to perform a resource checking operation during the given dispatch cycle to generate, for the given candidate sequence of decoded instructions, resource conflict information used to determine whether a resource conflict would occur. Resource conflict information cache storage is provided to maintain, for one or more sequences of decoded instructions, associated resource conflict information. In the event that the given candidate sequence matches one of the sequences for which associated resource conflict information is cached, the dispatch circuitry employs the associated cached resource conflict information to determine whether a resource conflict would occur, instead of invoking the resource checking circuitry to perform the resource checking operation.