Is it possible to have a partial Achilles tendon tear? If so, what would the symptoms be?

Yes. You can have a split or longitudinal tear or a tear that does not completely disrupt the full width of the tendon. Pain, swelling, weakness in plantarflexion and sometimes a palpable dell are findings.
Pain. Pain over the bottom end of the achilles, maybe difficulty pushing your foot down (like stepping on the gas).