Well, my dear friend, it is a very serious matter indeed to ignore the loving and merciful Grace of God in His offer of salvation (Mark 16:16; 1 John 5:12,13; John 3:36). God is indeed longsuffering and has reached out in mercy with a loving call to repentance and salvation to all mankind over the years; but, clearly God has made it known in the Scriptures that He will not continually entreat man to repent and come to Him for mercy beyond a certain point. Thus, the quick answer to your question would be “Yes” in the sense that God will not forever strive with those who stubbornly refuse His Grace and prefer carnality to godliness. Genesis 6:3 indicates that in the days of Noah, God graciously allowed 120 years for man to heed the warnings of Noah about the coming flood, which was the judgment of God for the wickedness of the world in those days; but man did not repent, so all but Noah and his family were drowned in the flood when it came (see also 1 Peter 3:18-20).

Now, I believe the verse you have in mind in your question is found in Romans 1:28 where we read: “And even as they did not like to retain God in their knowledge, God gave them over to a reprobate mind, to do those things which are not convenient…”. In this verse, God is clearly talking about those who from the beginning of man’s history to the present time did not honor the holy God despite all the evidence in creation that He is the divine creator. Therefore, for those who frankly do not want to submit to God, He has given these over to a reprobate mind (or you might say, God has allowed them to go their own way. (Reprobate in this verse, I believe, refers to a worthless, carnal mind which heedless of God’s holiness and devoid of any consciousness of sin,  desires and even enjoys wickedness.  Now, we must remember that we are all sinners (Romans 3:23), and we were all destined to judgment (Romans 6:23; Hebrews 9:27; James 2:10). I must stress here that we are not the judge of these things, but rather God is, and so we do want to continue to give the Gospel to the lost.  But, praise God, some of us do at length heed God’s warnings of judgment, and do repent of our sins, and do at length come to Christ.  But, for those who truly prefer life without God and do scornfully reject God’s authority, I believe that these are the ones who are turned over to their vile affections and wickedness, and do lose the capacity to see the truth, and thus are lost and destined to judgment (Romans 1:24-32).  Wm. MacDonald in his Bible Commentary on Romans 1:21 says: “Those who don’t want to see lose the capacity to see.” Since creation, God’s divine attributes and holiness has been made quite evident in nature, which is all around everyone, and of course, since Moses, we all can know about God through the written Word so that all the heathen are without excuse. And, as to the warnings of judgment for sins, the Bible is replete with those.  Some examples would be found in verses such as Genesis 6:3; Hebrews 10: 26-29; Acts 17:30,31; 2 Peter 3:3-11.  

Now, I do want to restate that God alone is the judge of the heart, so we as believers should continue to present the Gospel to those around us who may not know Christ. God alone knows who will receive the Gospel and who will not, so the Gospel should go forth to the world. John 12:32 says: “And I, if I be lifted up from the earth, will draw all men unto Me.” Of course, when the Lord refers to being lifted up, He is referring to His death on the cross, where He shed His precious blood in payment for the fearful debt of our sins; but now, what does the Lord mean when He says that He will “draw all men to Me”? I do not believe that Christ meant that He would bring every person in the world without exception to Himself in salvation, but rather that He would draw people from all over the world, some from every tribe and nation to Himself.  I believe this is important to recognize given your question, because there are many in this world who, despite God’s love and mercy, simply choose to go their own way.  These are not saved, but go on to judgment. In John 6:44 we read: “No man can come to Me, except the Father which hath sent Me draw him: and I will raise him up at the last day.” Those who do repent and put their trust and faith in Christ Jesus for salvation are those who were chosen of God before the foundation of the world (Ephesians 1:4). In John 3:16, it is only those “whosoever believeth in Him” who receive eternal life or salvation. Thus, when Christ died on the cross, he paid for all of our sins for all time, but salvation is only for those who repent and believe on Him. And then, we know that God has called those whom He foreknew, and that of all whom He has called, none will be lost (John 6:37-39). As we read in 2 Peter 3:9, God is in fact long suffering, and not willing that any should perish.  This is why the final judgment has not come upon the world yet. God alone foreknows those who will in fact come to Him.  Our gracious Lord is delaying judgment so that all the world has an ample opportunity to believe and repent.  Those who refuse God’s grace and mercy, who spurn the thought of the coming judgment, and who continue to perpetrate all manner of ungodliness, and encourage others to do likewise, are those whose hearts are allowed to go their own way, or to become reprobate, with all their vile affections.  (SF)  (581.1)