Can I rely on the promises God makes in the Bible or not?
The answer depends on what you mean, and what you mean depends on what your understanding is of "God's promises in the Bible".
There's only one promise you need to rely on, and that is that Jesus Christ is real, that He came into world to bear the sins of the whole world in Himself (and guess what, that's because we all indeed have sinned and come short of the glorious standard of God, as Paul stated in Romans 3:23-25).
His promise is that all who believe in Him will not perish, but have eternal life (John 3:14-18).
What does this mean, "will not perish, but have eternal life"? It means that Adam sinned, and died spiritually (and would therefore eventually die physically) when he disbelieved God, who said, "..For in the day that you eat of it you shall surely die." (Genesis 2:17).
Only once you understand how real that is, and how Adam's death came to all mankind, then you will understand what it means when Jesus' apostle Peter taught that God is "not willing that any of us should perish, but that all of us should come to repentance." (2 Peter 3:9) .. and you will also understand why Jesus said,
John 3
14 "But even as Moses lifted up the serpent in the wilderness, even so must the Son of man be lifted up,
15 so that whosoever believes in Him should not perish, but have everlasting life.
16 For God so loved the world that He gave His only-begotten Son, that whoever believes in Him should not perish but have everlasting life.
17 For God did not send His Son into the world to condemn the world, but so that the world might be saved through Him.
18 He who believes on Him is not condemned, but he who does not believe is condemned already, because he has not believed in the name of the only-begotten Son of God.
19 And this is the condemnation, that the Light has come into the world, and men loved darkness rather than the Light, because their deeds were evil."
You have to understand how real the above is before you understand what promise of God you need most, and need to rely on most.
Aside from that, Jesus never promised His disciples or those who believe in Him that we are going to be devoid of the same problems everyone born into the world faces - but we have a comforter who both comforts us and gives us peace in the most trying of circmstances - the Holy Spirit of Christ.