No, nested forms are forbidden.
This is expressed in the HTML 4.01 DTDs as:
<!ELEMENT FORM - - (%block;|SCRIPT)+ -(FORM) -- interactive form -->
— http://www.w3.org/TR/html4/interact/forms.html#h-17.3
This means A FORM has a mandatory start tag, mandatory end tag and can contain anything in %block or SCRIPT, except other FORMs.
XML DTDs aren't as expressive as SGML DTDs so in XHTML this rule is specified only in the human readable text of the specification:
form must not contain other form elements.
— http://www.w3.org/TR/xhtml1/#prohibitions
HTML 5 isn't an SGML application and doesn't have an official machine readable description of the language. It also expresses this rule in text:
Content model:
Flow content, but with no form element descendants.
— http://www.w3.org/TR/html5/forms.html#the-form-element
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