The short answer to this question is yes, a christian can and does struggle with sin. In fact, I would go so far to say that if one does not struggle with sin in this life then he or she is not a christian. The reason I say this is because the more holy one becomes the more they hate their sin and the sins they commit even if they have grown in their walk with Christ. But unbelievers do not struggle with sin because they do not care about whether or not they violate God's law. Thus, I believe, with the majority of Reformed Christians that Christians in this life always struggle with sin.
The main biblical text that justifies this position is Romans 7:14-25. In this blog post I will offer three of the strongest reasons for thinking that the person in Romans 7 is Paul, a believer, rather than a unregenerate Jew struggling with the law.
Reason 1: The person in Romans 7 relies and gives thanks to Christ and then continues to struggle.
Romans 7:25 - 8:1 25 Thanks be to God through Jesus Christ our Lord! So then, I myself serve the law of God with my mind, but with my flesh I serve the law of sin.
A unregenerate Jew does not have this sort of relation with Christ, rather only a believer has this sort of relation with Christ, a relation that is such that one gives thanks and relies on Christ in the midst of their struggles.
Reason 2: The person in Romans 7 is speaking in the first person and in the present tense.
This makes a good case that if this person is speaking in the first person and in the present tense that this is in fact Paul and that this is his normative experience as a believer.
Reason 3: The person in Romans 7 delights in the law of God and wants to do the good.
These two characteristics are not true of unbelievers but only believers.
For John writes of unbelievers:
John 3:19 19 And this is the judgment: the light has come into the world, and people loved the darkness rather than the light because their deeds were evil.
But Paul writes of the person in Romans 7:
Romans 7:15-18 15 I do not understand my own actions. For I do not do what I want, but I do the very thing I hate. 16 Now if I do what I do not want, I agree with the law, that it is good. 17 So now it is no longer I who do it, but sin that dwells within me. 18 For I know that nothing good dwells in me, that is, in my flesh. For I have the desire to do what is right, but not the ability to carry it out.
And again Paul Writes in Romans 7:
Romans 7:22 22 For I delight in the law of God, in my inner being,
It is pleasing to God that we love and delight in his law, but unbelievers cannot please God as Paul writes in the very next chapter:
Romans 8:7-8 7 For the mind that is set on the flesh is hostile to God, for it does not submit to God's law; indeed, it cannot. 8 Those who are in the flesh cannot please God.
Thus, given these reasons it seems that it is most reasonable to think that Paul is speaking of himself in Romans 7 and thereby showing that this is the regular life of the Christian.