Absolutely. Before COVID-19 ramped up, we noticed a remarkable number of patients diagnosed with both influenza and strep throat this season.
Also note--there are two types of coronavirus (229E and OC43) which cause the common cold. There is no cross-immunity for the two cold viruses. In other words, you potentially could get over a cold and come down with another, or be infected with both at once. OC43 is the most common.
Nothing like mixing the genetics of two virii strains with potential for a hybrid to be hatched.