Having some suspension travel is always nice, but what disadvantages are there for the bikes handling as the travel increases? This question applies to both to full and front suspension bikes.
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Although this isn't explicitly part of your question, I'll go ahead and throw it in as it's one of the most important factors to consider in terms of increased suspension if you plan on pedaling your bike- the basic principle behind propelling a bike is to convert a mostly downward force (pedaling) into forward momentum (the drivetrain turning the back wheel which in turn moves you and the bike). The more suspension you add, the more difficult it becomes to transfer the energy of pedal strokes into the drivetrain. The better circles you spin the less you'll suffer from this, but the fact remains that even with clipless pedals the majority of the power input to the drivetrain is coming from your glute, quad, and calf pushing down on the pedal during the powerstroke. Extra suspension soaks this up- it's like running on a trampoline.
Now that that's out of the way, let's talk about how additional suspension affects handling... after we briefly cover the benefits of adding suspension travel. Additional suspension will not only smooth out bigger, abrupt bumps (rocks, roots, your riding buddy that crashed in front of you, etc), but it will also smooth out more gradual terrain changes at higher speeds. Additional suspension basically allows the bike to track better and for the rider to maintain better control of the bike over rougher terrain and at speed. More suspension also helps the bike and the rider absorb bigger drops. Combined with a slacker headtube angle and longer wheelbase (all 3 tend to come as a package) you're also given a larger margin for error; mistakes that may have sent you over the bars on a cross country bike may be recoverable on a bike designed for bigger hits. These benefits come with tradeoffs, however.
Additional suspension will cause "muted" handling- that is that the bike won't feel as sharp or as connected to the trail. It will not turn as quickly or as precisely. This is due to a multitude of factors that come along with longer travel bikes:
I'm sure you've noticed that I've talked a lot about other factors beyond purely the travel of a bike's suspension alone, but it is important to take into consideration the whole package as all of the factors mentioned here work in symphony to create the ride quality of a bike as a whole, and each factor attributes to the ride quality in its own way- both positive and negative. You can't consider suspension travel alone because you can't find a bike where all else is equal beyond travel. You can't take a cross country bike and give it 8 inches of travel and call it a downhill bike anymore than you can take an F1 car and add 3 feet of suspension to it and call it a Baja truck- Each is a purpose built platform in nearly every aspect.
Regarding how increased suspension adversely effects handling. The first consideration is that to accommodate the increased travel you must make the bike taller so that the pedal/chainrings don't hit the ground as you utilize the increased travel. This results in higher center of gravity and the accompanying adverse handling changes.
Second; assuming there is an optimal suspension travel for the type of riding you are doing and you are asking, what are likly to be the advers handling effects of increasing travel? You also need to make the assumption that you will attempt to tune the suspension so that is used. Effects; Excessive loss of power to the suspension, sluggish dampened handling, difficult to tune and setup, excessive movement over terrain which can decrease stability when cornering.
This is not really an easy question to answer. To some extent you just keep getting more of the beneficial characteristics of suspension until them become detriments.
Suspension isn't something that can broken down simply, but let's try. First, you're asking about two different scenarios: front suspension and full suspension. Both setups can range from as little as 80mm on a hard-tail dirt jumper to 215mm on a full downhill/freeride bike.
Now, obviously, you're not going to want to grab your downhill bike to become the world's best XC racer. Why? For one, all bikes' geometry is geared around the purpose of the bike and how it's suspension aids that. Bikes with more suspension tend to have lower BB heights and longer wheelbases to aid in keeping the rider on the ground while traveling through the rough stuff. Shorter travel bikes usually have a steeper head angle to allow for greater steering: think of the XC racer weaving through tight trees.
A few of the disadvantages of increasing suspension:
After all that, there are lots of benefits to more suspension (probably should ask a separate question for that). It really comes down to the rider, the terrain, and what the rider wants to get out of it. Want a super smooth ride and don't mind pedaling? Then 6" of travel could work. Want to boost every lip and root, but still manage to rip some good speed on the flats? Maybe a 4" or 120mm trail bike with full suspension or hard tail is more for you.