Yes, it use to be that the senior apostle would be the next President of the church.  This happened with Brigham Young,  John Taylor, Wilford Woodruff, and Lorenzo Snow. They were all the Senior Apostle, and they all became president of the church.  But something happened on March 31, 1900 to change this.

It turns out that Brigham Young, in 1855, ordained his 11 year old son,John Willard Young , as an Apostle.  You can speculate all you want on why he would do this, but to me it looks like he was laying the path for a Young family dynasty of church leadership.

John Young served as a counselor to his father for several years, and as a “Counselor” to the Quorum of the 12, but he was never in the Q12.  He live a lot of time in New York City, and made some bad investments. He was accused by Joseph F. Smith of using church funds to support a rather lavish lifestyle, and the Q12 was looking to release him as a counselor when he resigned.

Nine years later, when it became apparent that John was the 2nd most senior apostle behind Lorenzo Snow, the First Presidency, fearing a John Young presidency would be a disaster for the church, changed  the rules of succession from senior apostle to senior member of the Q12.  And it has been like that ever since.

(For the most part this is just semantics, as I know of no other non Q12 Apostle that was even close to being the senior apostle, and the senior apostle has been the senior member of the Q12 ever since then, but rules are rules!)

For a more thorough treatment of this and other changes to the succession rules, see this Interview with Greg Prince  on the Gospel Tangents podcast.