Solubility of compound in acid
Yes, you've described a property of amines. When protonated by an acid (e.g. $\ce{HCl}$), an amine is converted into into a water soluble salt. This is actually a thing to do in pharmaceuticals to increase the amount of a drug that can be dissolved in water and be taken up by the body. The next time you have the chance take a close look at the chemical names on a drug label and theres a good chance you might see "hydrochloride."

Just as something can be protonated, it can be deprotonated. For the hydrochloride salt of an amine, $\ce{NaHCO3}$ should be basic enough to deprotonate it, and cause it to precipitate out of solution, as it would no longer be a water-soluble salt.