You are right in your assumption that Christian do get sick.  Let’s look at a few examples of believers being sick in the Bible. 

Matthew 8:14-15, “And when Jesus was come into Peter’s house, he saw his wife’s mother laid, and sick of a fever. And he touched her hand, and the fever left her: and she arose, and ministered unto them.”  We are not told exactly what this sickness was, but the Lord healed Peter’s mother-in-law of that infirmity that caused her to be bed-ridden.

1 Timothy 5:23, “Drink no longer water, but use a little wine for thy stomach’s sake and thine often infirmities.”  The Apostle Paul is writing to Timothy who seemingly was sickly and often had infirmities.  He encouraged Timothy to use wine medicinally to treat his stomach ailment.

Writing of Epaphroditus, Paul wrote, “For he longed after you all, and was full of heaviness, because that ye had heard that he had been sick. For indeed he was sick nigh unto death: but God had mercy on him; and not on him only, but on me also, lest I should have sorrow upon sorrow. I sent him therefore the more carefully, that, when ye see him again, ye may rejoice, and that I may be the less sorrowful. Receive him therefore in the Lord with all gladness; and hold such in reputation: Because for the work of Christ he was nigh unto death, not regarding his life, to supply your lack of service toward me.”  (Philippians 2:26-30).  Epaphroditus had evidently risked his life and health to minister to the Lord.  We are not told his sickness, but we do know it was serious for it was “night unto death.”

Some have taught that believers only get sick when there is sin in their lives.  While some behaviors do tend to cause bad health, like smoking or drinking, we cannot really say that a person’s illness is always related to some sin they have committed.  Let’s consider John 9:1-3 which tells us, “And as Jesus passed by, he saw a man which was blind from his birth. And his disciples asked him, saying, Master, who did sin, this man, or his parents, that he was born blind? Jesus answered, Neither hath this man sinned, nor his parents: but that the works of God should be made manifest in him.”  The Lord’s disciples upon seeing this man who was blind from birth, assumed that either he or his parents has sinned, thus resulting in his blindness.  It was quite foolish of them to think that possibly this man who was born blind had done some evil that caused his blindness.  Of course, the Lord reassures him that this man’s blindness was not a direct result of the behavior of him or his parents.  The Lord had allowed this blindness so that He might heal him and glorify God. (Read verses 4-7.)

We see that not all sickness is related to our individual sins, however some sicknesses are indeed related to our behavior and activities. Galatians 6:7 instructs us, “Be not deceived; God is not mocked: for whatsoever a man soweth, that shall he also reap.”  Bad habits can lead to bad health. 

Sometimes, the Lord may see fit to cause us to be sick so that we might learn lessons from it.  2 Corinthians 12:7-9 says, “And lest I should be exalted above measure through the abundance of the revelations, there was given to me a thorn in the flesh, the messenger of Satan to buffet me, lest I should be exalted above measure. For this thing I besought the Lord thrice, that it might depart from me. And he said unto me, My grace is sufficient for thee: for my strength is made perfect in weakness. Most gladly therefore will I rather glory in my infirmities, that the power of Christ may rest upon me.”  The Lord allowed this infirmity to come upon Paul to keep him humble.  Paul learned a priceless lesson from this.  He learned that God’s mercy was all he really needed. 

Christians are not immune to sicknesses and suffering. God allows sicknesses on Christians for various reasons.  Some are related to ours sins while others are not. Some sicknesses are allowed to test and develop our faith, to keep us close to God, and experience Him through our weakness.  It also gives us a stronger desire and expectation for our redeemed body in heaven. Although we may not fully understand the reason behind our sicknesses and suffering, we can be sure that God works for the good of those who love Him. All we need to do is to love God with all our heart and our mind and leave the rest up to God. 

While the Lord allowed His people to suffer terribly at the hands of the Babylonians, He reassures them in Jeremiah 29:10-11, “For thus says the Lord, ‘When seventy years have been completed for Babylon, I will visit you and fulfill My good word to you, to bring you back to this place. For I know the plans that I have for you,’ declares the Lord, ‘plans for welfare and not for calamity to give you a future and a hope.” (NASB) Even when we do not realize it, the Lord is working on our behalf (even through our sicknesses) to produce a “future and a hope” for those who trust in Him.  (436.2)