My favorite Bible verse that addresses this is in Proverbs:
It is the glory of God to conceal a matter;
to search out a matter is the glory of kings. (Proverbs 25:2, NIV)
Close on its heals is Jeremiah's promise:
10 This is what the Lord says: “When seventy years are completed for
Babylon, I will come to you and fulfill my good promise to bring you
back to this place. 11 For I know the plans I have for you,” declares
the Lord, “plans to prosper you and not to harm you, plans to give you
hope and a future. 12 Then you will call on me and come and pray to
me, and I will listen to you. 13 You will seek me and find me when
you seek me with all your heart. 14 I will be found by you,” declares
the Lord, “and will bring you back from captivity. I will gather you
from all the nations and places where I have banished you,” declares
the Lord, “and will bring you back to the place from which I carried
you into exile.” (Jeremiah 29:10-14)
After Jesus heals a blind man, the Pharisees are angered when the man claims that Jesus must be from God because he could give sight to the blind and so they throw him out of the synagogue.
35 Jesus heard that they had thrown him out, and when he found him, he
said, “Do you believe in the Son of Man?”
36 “Who is he, sir?” the man asked. “Tell me so that I may believe in
37 Jesus said, “You have now seen him; in fact, he is the one speaking
38 Then the man said, “Lord, I believe,” and he worshiped him.
39 Jesus said, “For judgment I have come into this world, so that
the blind will see and those who see will become blind.”
40 Some Pharisees who were with him heard him say this and asked,
“What? Are we blind too?”
41 Jesus said, “If you were blind, you would not be guilty of sin; but
now that you claim you can see, your guilt remains.
Not only does God hide, he has multiple reasons for hiding. One is due to our sin (a negative one). Another is to challenge us to find him. However, the most mysterious reason is that hiding brings God glory. Humans pursue fame, and fame requires that you be seen, your name be repeated, your deeds discussed. Fame to us is about being seen, not hidden. That God can exhibit his glory through hiding is a paradox.
@Zanarkand in his answer mentions Job. While Job was pouring out his complaint, he said this:
When he passes me, I cannot see him;
when he goes by, I cannot perceive him. (Job 9:11)
During these chapters when Job claims he could not see God, he spoke prophecies that revealed mysteries about the coming savior. It took me over a year of concerted study to find them. They were as hidden from me as they were from Job, and I had the benefit of commentaries and the knowledge that Job's words were part of Holy Writ. God's hiddenness and his presence are not incompatible.