Does the Bible teach the existence of guardian angels?
A short answer is yes. The catch is that sometimes short answers lead to conclusions that may not be justified. For instance, the idea that every child of God is protected from harm by a personal guardian angel, without a moment of inattention, is clearly mistaken.
On the other hand, angels are indeed assigned guard duty. Everybody is acquainted with Hebrews 1:14: “Are not all angels ministering spirits sent to serve those who will inherit salvation?” The Old Testament speaks frequently of special assignments. See Abraham's assurance to his servant—whom he had sent eastward to find a bride for Isaac—that the Lord would send His angel with him to make the journey a success (Gen. 24:7). That angel was the servant's guide and protector until the journey's end. See also Exodus 33:2 and similar references to the journey to the Promised Land.
The New Testament is equally rich in references to duties performed by angels on behalf of the Lord's servants. Read the account of Peter's escape from jail (Acts 12); you can check a concordance for other references to angels. My favorites relate the story of the angels in white uniforms who rolled away the stone at the tomb, showed the women that the grave was empty, and scared the guards half to death.
I personally do indeed believe in the existence of guardian angels. On unforgettable occasions I have experienced their protection.