Yes, they can. Christians can suffer from bouts of sadness, unhappiness even depression. In fact, the more spiritual you become and less carnal, the more cause you will need to reach out to the Lord. We can become sad, unhappy and depressed because we begin to see the carnal world for what it is. There are many Scriptures that share such stories about Christians and Old Testament Hebrews i.e. Mosses, Job, Jonah, Jeremiah who were confronted with anxiety, unhappiness and even depression. Both Paul and Peter in the New Testament surely felt all those.
Remember ~ "Come to me all of you who are weary and burdened, and I
will give you rest. Take my yoke upon you and learn me; for I am gentle
and humble in heart, and you will find rest for your souls. For my yoke
is easy and my burden is light." Matthew 11:28-30