A total of 215 cases of COVID-19 have been reported in San Pedro Town, of which only one has officially been reported as ‘recovered’. The latest updates on the island are showing a significant decline in positive cases compared to weeks ago. This raises the question of whether the Ministry of Health is testing island residents. The MOH clarified that concern, stating that swabbing for samples continues to take place, but not as many as before.

The Director of Health Services Dr. Marvin Manzanero said that cases on the island are expected to start decreasing as it has been almost a month since cases spiked. He attributed the recent decline in positive cases to the flattening of the curve as the alleged peak of the disease is beginning to drop. He reminded asymptomatic patients to voluntarily quarantine and to stay away from other persons to mitigate any potential transmission. On the island, people have been complaining that when visiting the Mar de Tumbo flu clinic for testing, they are turned back. Manzanero said that they cannot test every single person and that at the moment only symptomatic, pregnant women and high-risk patients are being prioritized.

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Why can't they test everyone who presents themselves for testing?

What would be the purpose of testing anyone who wanted. If they don't have symptoms or just mild systems, they would only be told to self quarantine. It would be a waste of tests.

"He reminded asymptomatic patients to voluntarily quarantine and to stay away from other persons to mitigate any potential transmission."
I'm pretty sure "asymptomatic" means no symptoms, so, how would an "asymptomatic" know that they are infected?

There are many more recovered than the MoH is listing...

Antibody testing in the United States revealed that for every person with symptoms, there are 9 people who did not get sick and in Spain, for every registered patient there were 15 asymptomatic:
https://elpais.com/sociedad/2020-04-07/mas-del-90-de-contagios-estan-ocultos.html

I think the logic is if you were around anyone that tested positive, it would be prudent to self quarantine to see if you developed symptoms.

Exactly. And lots of people I know were exposed and had symptoms and isolated who were not included in the "count." If the system says we have 200 cases it is logical to assume that we have more cases. It is also logical to think that there are many who likely contracted the virus and are showing no symptoms at all.