The performance of a great variety of electronic devices—ranging from semiconductor transistors to superconducting qubits—is hampered by low-frequency noise with spectra proportional to 1/f. The ubiquity and negative impact of 1/f noise has motivated intensive research into its cause, and it is now believed to originate from a bath of fluctuating two-level defect states (TLSs) embedded in the material. This phenomenon is commonly described by the long-established standard tunnelling model (STM) of independent TLS. A key prediction of STM is that the noise should vanish at low temperatures. Here we report measurements on superconducting microresonators over previously unattainable, very long time scales that show an increase in 1/f noise at low temperatures and low microwave power, contrary to the STM. We propose a new generalised tunnelling model that includes significant interaction between multiple TLSs, which fully describes these observations, as well as recent studies of individual TLS lifetimes in superconducting qubits.