Some of them we can, and others we cannot. Many of the promises given to the Israelites are simply promises based on God's care for His children and speak of His own faithfulness and goodness. Those statements and promises are for all time. But there are other promises that are specific to the Mosaic Covenant. Those we CANNOT claim because they were given to Israel and had a very specific promise attached to them that pertained to the LAND that was promised to Abraham as well as the blessing that would attend their obedience to the Law.
As the Apostle Paul tells us, "We are not under law" (Romans 6:14) which means we are not involved in the covenant that God made with Israel through Moses. In fact, even in the Old Testament prophetic writings, God told Israel that He was going to make a new covenant with them — which is the covenant of grace we now have through Jesus Christ. (See Jeremiah 31:31) So, technically even the Jews of Israel can no longer claim the promises of the Mosaic Covenant. God offered them a new covenant through His Son. The fact that they rejected His Son and the New Covenant doesn't mean the Old Covenant is still in force. As the writer of Hebrews says, it is "obsolete" and "ready to vanish away." (Hebrews 8:13)