When does a message queue add more complexity than benefit

We're thinking about adding a message queue to our setup, probably RabbitMQ or Kafka, to split up some of our services. But I've been wondering about the downsides. When does the extra work to run it, plus the need for new monitoring and figuring out queue problems, become more trouble than it's worth compared to asynchronous communication and decoupling? I'm looking for actual stories where sticking with synchronous communication was actually the smarter move.