Yes — and it is a common cause
When a gutter overflows, water runs straight down the wall instead of away through the downpipe. Brickwork that is repeatedly saturated stays wet, and that moisture can work its way inward.
What it looks like
Externally you may see staining, moss on the wall, or spalling brick. Internally, damp from a gutter problem often shows as a patch high on a wall or across a ceiling corner, worse after prolonged rain.
Check the outside first
Before treating damp internally, it is worth confirming that rainwater is being carried away as it should be. An inspection of the guttering and downpipes will show whether the source is outside.