Can chronic chest infection exist when one feels that it is present but there is no cough, phlegm, shortness of breath and xrays are negative?

Infection not likely. A person with no shortness of breath, cough, phlegm, or chest x-ray abnormalities should not have any chronic lung infection. If he just inhaled infectious germs, he may get symptoms (of acute infection, not chronic) a few days or weeks later. If a person "feels" there is something unusual in his chest, despite no infection symptoms, he should be evaluated by a primary care doctor or lung doctor.
Unlikely. While it is possible to have an asymptomatic infection- usually chest infection cause some symptoms, such as a cough or shortness of breath.
Other possibilities. If there is an infection of the airways, you would expect to see some of the symptoms you describe. If pneumonia is present, your cxr would reveal it. But, you may be experiencing bronchospasm associated with airway inflammation, depending on your specific symptoms and physical exam findings so see your doctor for diagnosis and treatment. Good luck!