This ‘can't get rid of this cold' thing is a problem that hits most people sometime in their lives.
A cold virus will usually go through a cyclic battle with your immune system in a few hours...or days if you have a bad influenza. Often there follows a residual bacteriological infection--sometimes termed a secondary infection.
The first type of ‘germ' will not respond to antibiotics, the second hopefully will.
Doctors tend to prescribe antibiotics--even during the viral stage, cos they know that the odd are that you will get the secondary. I usually try to avoid the use of such sledge-hammer remedies, but I do like to have the antibiotic ready. Especially while away from base.
Back to your specific case. One question. Do you have children in the house? Sometimes one can get back-to-back viral infections, leaving you with the impression that you are just not getting better from the first one. Kids do this to their parents.
One of my kid's ‘specialty subjects' is the immune system. He had such grand ideas about not allowing his lot to be rushed onto antibiotics...until he came here. Both parents have been knocked over by repeated infections brought home by their kids.
As an aside for general interest. This information came from one of my neighbors, a PhD teaching a relevant subject.
Many infections are caused by the Mexican children in day care here being mixed with the local kids. The Mexican worker go home each night allowing a much more interesting mixture of viruses. Relief nursing staff come here for a six week stint, and often spend half of the time laid out with severe infections.
In this modern world, where there are few boundaries for this type of population mixing, we are all likely to go through a really rough period now and then.