Can you have nested forms? Simply no, Nested form won't work.
Docs Says
In Angular, forms can be nested. This means that the outer form is
valid when all of the child forms are valid as well. However, browsers
do not allow nesting of elements, so Angular provides the
ngForm directive which behaves identically to but can be
nested. This allows you to have nested forms, which is very useful
when using Angular validation directives in forms that are dynamically
generated using the ngRepeat directive
But unfortunately you could not use the ng-submit form, that will call the parent ng-submit method on click of any inner form.
Example Plunkr
Workaround for this would be don't use ng-submit directive for the inner nested from. You should use ng-click on button and change type of button to button from submit.
Markup
<form name="add_foobar_form" novalidate ng-submit="addFoobar('foobar')">
<ng-form name="send_contact_form" novalidate>
<input type="text" ng-model="hello.bye">
<input type="button" value="sendmall" ng-click="sendContact('hello')">
</ng-form>
<input type="text" ng-model="foobar.go">
<input type="submit" value="sendall">
</form>
Workaround Plunkr