Yes, it is common for true believers to experience seasons of doubt, as assurance is not of the essence of saving faith. While we should not remain content in our uncertainty, such doubts do not negate one's standing in Christ, as the Lord provides grace and support to keep his people from sinking into despair.
It is a tender truth that even those who are truly in Christ may struggle with heavy hearts and waves of uncertainty. Our confessions remind us that assurance is not identical to faith itself; one may possess a genuine, saving interest in the Savior while lacking the full, felt confidence of that relationship. Whether through the trials of life, the weight of sin, or the mystery of God's providence, believers may find their peace shaken. However, we are never left in a state of hopelessness, for as the 1689 LBCF Ch.18 §4 and the Canons of Dort Head 4 Art.11 assure us, even in the darkness, the 'seed of God' remains within us. We are supported by the Spirit so that we do not sink into total despair, but are sustained by the life of faith until that assurance is revived in due time.
When we find ourselves doubting, we must remember that our salvation rests not upon the strength of our faith, but upon the object of our faith: Jesus Christ. Even the apostle Thomas, who doubted the resurrection, was met by the risen Lord with grace rather than condemnation. As recorded in John 20:27, Jesus invited him to see the evidence of His wounds, calling him to move from faithlessness to belief. Like Thomas, we are invited to look at the work of Christ. As Charles Spurgeon wisely noted in All of Grace, the way to resolve doubt is to look away from our own internal instability and fix our eyes upon the authority of God's Word and the sufficiency of Christ's sacrifice. We are accepted in Him, even when our own hearts feel uncertain of that fact.