This is an unfortunate problem that can sometimes occur after nose surgery
or with advancing age, as the delicate cartilage that supports the side walls of the nostrils is weakened. This is called nasal valve collapse, and there is a treatment for this condition. Before anything, however, you can test for yourself if this could be the problem, either by using the external nasal strips that hold open the nostrils, or by pulling gently outwards from the sides of your nose and seeing whether or not your symptoms improve. If your symptoms improve, then nasal valve collapse is more likely.
Regardless, you will need to speak with an ear-nose-and throat surgeon
, aka ENT) to determine how best to correct this condition, which may include surgery to bolster the support to the sides of your nostrils. Alternatively, you could just continue using the nasal tape to help with your symptoms. Most people prefer surgery for the fact that wearing tape on your nose every day is cosmetically sub optimal.
If you do visit with a surgeon who recommends surgical repair, please discuss with him or her how likely it is that you will need the surgery repeated. While nasal surgery is very delicate, excellent and permanent results are possible.