"Is today the eleventh or the twelfth?"
Translation:Je dnes jedenáctého nebo dvanáctého?
Yes, the nominative can also be used. However for a date you would typically have to add the month.
Dnes je jedenáctý listopad a zítra je dvanáctý.
...unless the month was mention in the preceding question.
Kolikátý je dnes listopad? Dvanáctý. (Dnes je dvanáctý.)
However, this genitive contruction is a specific construction used for dates.
Je dvanáctého srpna. Je prvního dubna.
The rule that "to be" goes with the nominative is not 100% reliable, it's just a general rule. Take, for example, a sentence like "This is Tom's" (i.e. this belongs to Tom) -> "Tohle je Toma" with "Tom" in the genitive. Similarly, dates are in genitive by default, unless something (a preposition) forces them to be in another case.
(And BTW, "This is Tom's" can also be expressed using the possessive adjective: "Tohle je Tomovo", where we use the nominative again.)