Is it possible that a Muslim commits sins but is still considered a Muslim?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.

 Some actions contradict Islam; however, they do not drive man out of the fold of Islam. Faith consists of many branches. On the authority of Abu Hurayrah the Prophet, sallallaahu ‘alayhi wasallam, said that faith consists of over seventy or over  sixty parts, the highest of which is testifying that there is no deity worthy of worship except Allaah The Almighty and the least of which is removing harm from roads. He added that bashfulness is a part of faith.  [Al-Bukhari and Muslim]

Faith  does not exist if specific parts do not exist, such as the Testimony of Faith where the person bears witness that there is no one worthy of worship except Allaah The Almighty; if a person does not acknowledge that there is no one worthy of worship except Allaah The Almighty, he is not a Muslim.

This also applies to a person who does not believe in the truthfulness of what the Messengers brought from Allaah The Almighty; to a person who ridicules all that is sacred in Sharee'ah such as the messengers, Divine books and angels; to a person who denies indisputably-established facts of the religion like the obligation of prayers and fasting and the prohibition of consuming alcohol and swine; to a person who abandons prayers. According to some scholars, the latter ruling is supported by authentic Hadeeths. It was reported that the Prophet, sallallaahu ‘alayhi wasallam, said that a person who abandons prayer is a disbeliever. [Ahmad, An-Nasaa’i and Ibn Maajah]

If some specific parts of faith do not exist it does not invalidate one’s faith; an adulterer or fornicator, a robber or someone who drinks alcohol are all considered Muslims as long as they do not deny that these acts are prohibited. However, their faith is considered to be incomplete. On the authority of Ibn ‘Abbaas the Prophet, sallallaahu ‘alayhi wasallam, said that a Muslim is not a believer at the time he robs, fornicates, commits adultery or drinks alcohol. [Al-Bukhari and Muslim] This Hadeeth may mean that such a person’s faith is incomplete or defective, that his heart does not fear Allaah The Almighty while he is committing such acts; that he is doing the actions of non-Muslims. The Hadeeth may also mean that all this will lead him to disbelief or that he will be a disbeliever if he thinks that any of these acts are lawful. By doing so, a person deserves to be punished if he does not repent, but he will not reside eternally in Hell. The Prophet, sallallaahu ‘alayhi wasallam, said that the person in whose heart there is the least amount of faith will not reside in Hell eternally. [Muslim Abu Daawood, At-Tirmithi and Ibn Maajah]

In brief, Muslims vary in their ranks; a person of the highest degree is the one who worships Allaah The Almighty as if he sees Him and consequently never commits any prohibited action. Some other Muslims may commit some minor sins such as taking quick glimpses at something that is unlawful or lying with the intention of joking. About these people, Allaah The Almighty Says (what means): {If you avoid the major sins which you are forbidden, We will remove from you your lesser sins and admit you to a noble entrance [into Paradise].} [Quran 4:31] 

Some other Muslims practise indecency and they commit major sins. Those people deserve to be punished because of their deeds, but Allaah The Almighty by His favor and great mercy may forgive them. Allaah The Almighty Says (what means): {Indeed, Allaah does not forgive association with Him, but He forgives what is less than that for whom He wills.} [Quran 4:48]

 He also may not forgive them, and if that happens they will be punished but this punishment will not be eternal.

On the authority of Abu Tharr the Prophet, sallallaahu ‘alayhi wasallam, said that if any person from his Ummah dies without associating others with Allaah The Almighty, he will enter Paradise even if he has committed adultery or has stolen. [Al-Bukhari and Muslim]

Likewise, you should know that all sins are forgiven by sincere repentance to Allaah The Almighty Who Says (what means): {Say, "O My servants who have transgressed against themselves [by sinning], do not despair of the mercy of Allaah. Indeed, Allaah forgives all sins. Indeed, it is He who is the Forgiving, the Merciful."} [Quran 39:53]

Allaah Knows best.