There are many things that could cause a cough like that, but the chances of it being serious are extremely small. I've known a lot of people who had a long-lasting cough that turned out to be just asthma, or allergies, or even walking pneumonia (which isn't as scary as it sounds). My daughter had a cough and wheezing that ended up being a combination of asthma and bronchial infection. My dad also had something that made him cough for over a month, but the doc said it would go away on its own, and it did. It is also possible for adults to develop allergies suddenly, so your grandparents could be right about that. Probably the only way to know for sure is to see a doctor.