"Some doubt that. Why invest in these sectors during a slumping economy, they ask?"

That construction would have obviated Porche's objection to an example I used in explaining that the terms "that" and "why" cannot be combine as immediately adjacent terms of a single sentence.

Adding that attributive phrase "they said," the relationship of the two sentences would have been made explicit.

Nomad your third post indicates a deft understanding of the phrase "doubt that."

However, the example sentence of your initial post ("Some people doubt that why ...") is not just weird, it is defective at the level of syntax rather grammar.

On that point, consider this simple modification:

Some doubt that. Why invest in these sectors during an economic slump?

The two sentences are correct in every respect, but when run together, as in the example, a problem arises. Why?

The terms, "that," a demonstrative term, a term that points something out, and "why," an interrogative term, a term that questions, logically cannot be combined without some intervening term--such is the clash between their functions.

The problem with first sentence is its use of the word "why." The sentence would read more intelligibly:

"Some people may have doubts that investing ..."

However, the thought would be conveyed more effectively:

"Some people doubt that investing ..."

If required to use the noun form of the phrase in question, I would use a stronger verb than "have:"

"Some people harbor doubts that ..."
That is to say, some hold guarded or barely expressed doubts.