Yes, they can have 'cardiovascular accidents/incidents', where their blood flow is not sufficient to take care of their vital organs, and so the result seems like a 'stroke' in a human.
The heart is a huge muscle which needs lots of Calcium to work. If the bird has been eating mostly seeds without Calcium supplementation (like a cuttle bone for budgies), and if there is no available Vitamin D3 to help make the Calcium work, then you can see 'stroke-like' activity.
