Yes, but it's very rare and hard (actually for most people, impossible) to individually verify.
Quran mentions various individuals other than prophets who received some form of inspiration from Allah, examples being Mothers of Prophet Moses and Prophet Jesus peace be upon them. Lady Maryam in particular saw Gabriel or Jibri'il.
In Sufi and Mystical tradition of Islam, it is also believed that angels may appear to Muslim mystics or very pious believers. However, not all (even most) of the claims to visiting or communicating with angels can be verified.
The 12th century prominent Muslim mystic, Muhye ed-Din Ibn Arabi, in particular has claimed to even have actually seen Arch Angel Gabriel! But he remains controversial among Muslim juridical orthodoxy (mostly for heretical appearance of some of his theological beliefs).
Shias in particular believe their Imams could communicate or see angels. Likewise prominent Shia mystics (who were almost all accomplished jurists too) also believe that seeing angels (even Archangels) is a possibility and there have been actually contemporary highly respected Shia mystics who are said to have seen angels: examples Ayatollah Seyyed Ali Ghadi, and Ayatollah Allama Muhammad-Hussain Tabataba'i.
Many Muslim mystics, both Shia or Sufi, also don't believe that communication even with Archangels is a defining characteristic of Prophets. What distinguishes Prophets is that they are inspired by Allah to establish a religion not that they were simply inspired by Allah or saw angels. But since truly God-inspired infallible saints and prophets can't contradict each other, there will be no true mystic to claim a new religion after Prophet Muhammad (pbuh) because the Prophet of Islam specifically stated that Islam would be the final religion.
However like I said, since such supernatural achievements are rare and difficult to verify, the default position is that such claims must not be taken seriously. In most cases the person making the claim could be an imposter or deluded or simply having had an exceptionally impressive dream vision that he interprets as a Divine inspiration, which could be the case if the person claiming the experience is a particularly imaginative, intelligent and/or ambitious individual.
Alternately, the claim could be simply a result of medical conditions or, even if truly a supernatural phenomenon, it could be the result of a demonic inspiration rather than Divine! Demonic inspiration is more likely if the person is of a particularly poor mental health or a very sinful life (like heavy drinking, sexual promiscuity, drugs, blasphemy etc).
So if the content of the claimed message or vision is clearly contradictory to teachings of Islam then it should be certainly dismissed. If it comes from people who have had serious psychological conditions or sinful lives it should be likewise dismissed. If there is claim of a personal intermediary like angels then the stakes would be even higher since encounter with supernatural persons is rarer than simply hearing or feeling some supernatural phenomena.
Therefore, as for the claim of your friend you may say to him that Divine inspiration is very rare and even if the case, you usually expect it from very pious people. You also say that otherwise, the experience could be simply a delusion. You may also hint to him how demons (jinns) can also inspire people. You can cite Surat an-Nas as Quranic basis for demonic inspiration. By raising these points you reasonably dismiss his claim and then simply move on!