We live within city limits.
Depending on where you live, the sewer system may be tied into the storm drain system. When you get enough water in the system, it can back up into the sewer line in your house
A sewer backflow valve can prevent waste water backups, particularly from a public sewer. But before we get into the details and benefits of a sewer backwater valve, let’s explain what backwater is. Sewer backflow is a term used in plumbing for unexpected and unwanted flow of water in reverse direction. Normal plumbing allows wastewater to flow from a home to the city sewer, but backflow is the exact opposite.
When backflow happens, there can be a serious health risk due to contaminated waste water entering your premises. Backflow occurs when the municipal drainage system or city sewer overflows, and sends water back through a sewer pipe into your home. Immediate proper clean-up is required to prevent risk of disease and further damage to property.
You might be experiencing what that article terms a "sewer surcharge", but on a smaller scale. If you do not have a backflow valve installed (and most homes do not), the sewage can back up into your house. Even if this is not the issue here, I would check to make sure you have one installed.