Non‐technical summary  Exposure to chronic intermittent hypoxia (CIH) leads to chronically elevated sympathetic nerve activity (SNA) and mean arterial pressure (MAP) with exaggerated rises in SNA and MAP to activation of peripheral chemoreceptors. We show that CIH leads to augmented rises in SNA by stimulation of reflexes from other sensory nerves, suggesting a global change in how the brain processes sensory information. Furthermore, the ability of glutamate in the rostral ventrolateral medulla to increase SNA and MAP is enhanced after CIH, providing a potential brain mechanism for CIH‐induced exaggeration of sympathetic reflexes. Paradoxically, exaggerated sympathetic reflexes were accompanied by normal rises in MAP. In agreement, activation of adrenergic vascular receptors yields blunted rises in MAP in rats after CIH. These data suggest that exposure to CIH facilitates rises in SNA, potentially by changes in the brainstem, which are buffered by a reduction in the ability of sympathetic nerves to raise MAP.