if someone has a serious medical problem and is receiving treatment, can that person buy insurance?

If so, will the insurance company cover the treatment costs?

Yes, normally they can. But, the condition they are being treated for won’t be covered. This is true with any private insurance company anywhere in the world.
A “pre-existing condition” that requires on-going treatment is a “guaranteed expense” to an insurance company. The only way they could afford to cover “guaranteed expenses” is by passing the cost on to the rest of their insureds in the form of higher premiums. When an insurance company is forced to do this, its insurance plans become less competitively priced against those of its competition to such a degree that only “high risk” insureds are willing to pay the premiums.

Insurance plans that do cover pre-existing conditions are normally government sponsored social-medicine schemes. In America, where “socialized-medicine” is a bad word, they have HIPPA and COBRA to provide coverage to people with health problems who can’t get coverage elsewhere. Coverage under COBRA, though, is extremely costly. Some large corporations also provide employees with “guaranteed acceptance” group insurance coverage that doesn’t exclude pre-existing conditions of any sort. The cost provides a large tax write-off for the company.

What does an insurance company do about pre-existing conditions the insured didn’t tell them about, or the insured actually didn’t know about, but, the company discovers after the policy gets issued?

a) If the insured lied, and the company learns that the insured lied, the coverage will be canceled and all claims will be refused - including even the legitimate claims.
b) If the insured seeks treatment for a pre-existing condition the insured was most likely unaware of, the condition wont be covered, but the coverage will continue.

That said, the good news is that the insurance companies are faced with a time-limit for finding and excluding pre-existing conditions that they haven’t already found and excluded.

Insurance companies have up to 24 months to find and exclude any pre-existing conditions you might have. Any pre-existing conditions that haven’t been found and excluded within the first 24 months the policy has been in force can no longer be excluded from coverage.

One of the biggest fears insurance companies have to face with each insurance application is that the applicant is taking out the insurance policy to cover a major medical expense that the applicant is expecting to incur soon and wants pass the expense on to the insurance company. The time exclusion is intended to make that less likely to happen.