Are all the promises given in Scripture applicable to all believers, or only some?
The short answer is no, not all promises in the Bible are applicable to believers. In fact, many promises in the Bible are not applicable to believers at all. When God makes a promises in His word, we must examine who He spoke to and what He said before we can know whether those promises will apply to all believers or to a specific group of believers.
All Scripture is useful for instruction and training, as Paul says, and therefore all Scripture has been written for our benefit. On the other hand, that doesn't mean every statement of Scripture is true for us. For example, God spoke specific promises to Abraham at several places in the book of Genesis. God promised to give Abraham a son, and to produce a nation from Abraham's seed, and to give Abraham a land and a name. These promises were given specifically to Abraham, and they are begin fulfilled through the Jewish people.
Proper use of scripture requires that we interpret those verses in their context, which means they were promises spoken to Abraham and to his descendants. God's promises included blessing many nations (Gen 22:18) through Abraham, so we know we may receive the fruit of those promises, but that doesn't mean the promises are spoken to us. Consequently, we cannot claim that the promises spoken to Abraham were intended for all believers.
Similarly, God spoke promises to the other patriarchs and to many kings of Israel including David, He spoke promises to Moses and the people of Israel, and He spoke promises to Israel's enemies. Are believers supposed to lay claim to all those promises? If you examine all the promises God has spoken in the Old Testament, there are many that we would never want to claim.
For example, God made promises of judgment to many peoples, usually to Israel's enemies. Many of these promises of judgment are found in the book of Isaiah, and after a careful study of that book, you would quickly conclude that God's promises are not necessarily intended to be applied to all believers (nor would you want them!).
Whenever we see a promise in Scripture, always ask the following questions: Who did the speaking? Was it God speaking directly or through a prophet or Apostle? Who was receiving the promise? Was the promise given to a certain individual or group (e.g., the Israelites or the Apostles)? Who would benefit from the promise (e.g., just the hearer or others as well)?
As you answer these questions, you will find the Scripture's focus either narrowing or expanding. When it narrows, as in the case of Abraham or promises for judgment, then we shouldn't try to expand them. When the promises expand, as in the case of Jesus' or Paul's teaching in the New Testament, then we should make an appropriate application for our own sake. Let the context of the Scriptures guide your interpretation.