More bulk means more mass to move, especially in the upper body. If your arms and shoulders are bulky, it can be hard to launch and retrieve them when punching. Violent movements with more extra mass requires more muscle in the lower body than in the upper.
Lower flexibility: Just like you can be bulky and flexible (to a point), you have to realize that extra muscle just makes that more difficult. Teaching your muscles how to contract can set a neural pattern that is hard to reverse, especially in intense situations. So if you count on using your flexibility in the fight, you’ll need to do a ton of stretching and relaxation exercises before the fight.
Shortened Reach: This is just a matter of mass. Your frame can only support so much weight. If you’re 5’8″ and bulky as hell, then you might be facing a fighter with a 6’5″ athletic frame.
Functional Strength: If your muscle came from resistance training with machines and isolated movements, then you’re strength isn’t as good as someone that trained their muscles throughout the body to contract as a complete kinetic chain from fingers to toes. For example, if you work with a medicine ball, you train the body as a whole to perform complex movements explosively. Seated cable curls might not be the best use of energy/time for your resistance training.
Extra Muscle: You can really wear your opponent out with your muscle, if you can hold him in a bad position. Most collegiate wrestlers look a tad bulkier than elite MMA fighters because muscular strength is more useful in grappling situations. Weight/mass is a big advantage in scrambles and other position changes.
Motor Unit Recruitment: If you train your muscles to contract with efficiency, then you can learn to activate more muscle fibers with better coordination. This means a faster, more explosive contraction. Think of it as refined muscle memory. A lot strength gains come from refined motor patterns, not just from growing bigger muscles.
A more refined muscular force gained from practicing a movement at full speed is almost like saying technique training. As a rule of thumb, any activity that adds muscle to your body should closely mimic the same positions and actions commonly found in MMA. That training should also activate the same energy systems (aerobic and anaerobic) used in a fight. You need to train explosively and repeatedly, for power and cardio, respectively.