Can I precede a noun with more than one determiner?

Is there a rule that a noun would take only one determiner at most?

For example, according to “Determiner” at EnglishLanguageGuide.com, both both and the are determiners. Can I write an expression like the both cats? Or do I have to use it in either the cats or both cats forms only?

Is this usage correct?

I saw two cats this morning. The both cats were very young.

Yes, more than one determiner can precede a noun, but they do so in a particular order. All, both and half come before articles, so your example would have to read I saw two cats this morning. Both the cats were very young (but in this case the can be omitted).

As noted in previous answer, “Both the...” is standard. However, in many cases of spoken (vs written) English, one finds “the both of them/you/us” being used where “both the” would be standard.

Merriam-Webster's Dictionary of English Usage has half a dozen paragraphs under both, sense 5, with examples including “the both of you” and “the both of them”. It concludes:

The expression appears to be an Americanism ... There is no reason you should avoid it if it is your normal idiom.

Note, Merriam-Webster's advice does not countenance saying “The both cats were...”, but would allow certain Americans to say “I saw two cats. The both of them were white with spots of gray.”