The Wikipedia article on Kshatriyas suggests possible rivalry between the two varnas due to ambiguity in their relative positions in the society.
In the period of the Brahmanas (800 BCE to 700 BCE) there was ambiguity in the position of the varna. In the Panchavimsha Brahmana (13,4,7), the Rajanya are placed first, followed by Brahmana then Vaishya. In Shatapatha Brahmana 22.214.171.124, the Kshatriya are placed second. In Shatapatha Brahmana 126.96.36.199 the order is—Brahmana, Vaishya, Rajanya, Shudra. The order of the brahmanical tradition—Brahmana, Kshatriya, Vaishya, Shudra—became fixed from the time of dharmasutras (450 BCE to 100 BCE). The kshatriya were often considered pre-eminent in Buddhist circles. Even among Hindu societies they were sometimes at rivalry with the Brahmins, but they generally acknowledged the superiority of the priestly class.
But are any real instances of this rivalry between the two classes mentioned in Hindu scripture?