I will admit right up front that I have done nothing more than anecdotal research on this topic.

So, then why are you presenting your opinion on the matter? Why does anyone do this?

News outlets are constantly telling their audience that there is too much sodium in our diet.

Your doctor will tell you the same thing. Because it's true. Increased intake of salt and fat intake is associated and causally linked with hypertension, which increases risk of heart attack and stroke, both of which carry the risk of dying.

If you dilute something into solution under pressure, like dietary sodium or fat globules in your blood, it increases blood pressure. It's basic fluid dynamics.

I remember reading that people with high blood pressure can have that exacerbated by too much salt. However, the salt did not cause the condition.

Yes, it did, or it was one of many factors that did. Which is why doctors prescribe things like the DASH diet, or low-fat, low-salt diets. The narrowing of the arteries by plaque build up can be made worse by increasing the pressure inside of said arteries, in addition to the fact that all of that salt is also associated with heart disease.

Can anyone tell me a problem directly caused by higher than recommended sodium intake?

Increased blood pressure, which is associated with a number of health problems, which is bad and is worsened by more salt, more fat, and stress.