Wolfram Language (Mathematica), 44 bytes

If[(s=FromRomanNumeral@#)<3999,s,F]~Check~F&

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returns the value for truthy and F for falsey

the limitation [1,3999] unfortunateley costs 16 bytes...
here is the code for any roman number

Wolfram Language (Mathematica), 28 bytes

Check[FromRomanNumeral@#,F]&

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the above code works for any number, not just [1,3999]