Electronic devices may be provided that contain wireless communication circuitry. The wireless communication circuitry may include radio-frequency transceiver circuitry coupled to antennas. An electronic device may include a baseband processor and other storage and processing circuitry that implements protocol stacks for handling multiple radio access technologies. The storage and processing circuitry may use the transceiver circuitry to convey data using a first radio access technology while periodically interrupting the conveying of the data to monitor a paging channel using a second radio access technology. In performing the paging channel monitoring operations, the storage and processing circuitry may enforce a time limit that ensures that operations using the first radio access technology are not disrupted more than desired.