Boredom sets in when we become tired of something, and usually this happens when we do it over and over until what we are doing loses its novelty or its purpose, or we meet a difficulty that we just don't feel like overcoming. So we usually say we're bored when what we are doing no longer entertains us or we are simply not willing to put in the effort to take it to the next level. It can happen with any activity, learning an instrument for example, or trying to take a sport to the next level. And we presume something will be boring if it is repetitive, seems dry and unentertaining or involves difficulty that just doesn't seem worth it.
Now, just as "one man's food is another man's poison", what excites one person and gets him enthusiastic can be killingly boring to another. What does it depend upon, mostly? On your interests and your abilities.
Now, getting to the Mass and the divine office, since they seem repetitive from the outside we easily get the impression they must be boring. But if they are or not depends precisely on your interests and abilities, and not on the repetitiveness. A pianist can practice the same arpeggios over and over, or a hockey player the same move for days on end, and far from boring them the repetition gets them more excited as they see their progress. So, in our prayer a lot depends on what we are willing to put into it, why we are doing it, what our motivation is, if we are just saying words or if we are contacting with God, adoring him, thanking him, entering into his mystery, discovering the person of Christ, asking for what we need, interceding for others, praying for those who don't know him, trying to find the grace to change ourselves and grow.
And then, as regards our abilities: we spontaneously think that prayer is for people who are "holier" than we are, people who find it easy, while we have to contend with tiredness and dryness, our faith seems so weak, so we figure we don't have what it takes. Here is where we have to remind ourselves of a great truth: in Baptism we received the gifts of Faith, Hope and Love. Baptism changes us, makes us new creatures in Christ, gives us the qualities and the ability to live a totally new life based on these three virtues, and this gives us the "raw talent" to be Christians, for Christ to mean something to us, and to be able to develop those gifts the way any person with raw natural talent develops his, by practice. If God didn't give us these gifts we would have every reason to give up.
There is something else. Every person who attempts to pray always runs into dryness, difficulty, which sometimes can last quite a while. Does he not get bored with prayer then? He certainly does get tempted to give up, but even that temptation is a way to progress in prayer, overcoming it is an opportunity to let God work, to allow him to purify us, and to thank him.
I've wandered a bit, but I hope these thoughts help.