On a firing range with your weapons already loaded, it might seem like the better choice for warfare would be the lever action - it can fire off those bullets significantly faster than the bolt action.

However, the disadvantages of the lever action comes when it times to reload - a lever action needs to have the bullets individually placed into it, one at a time. With bolt action weapons you can just use a magazine, so in the time it takes a lever action to reload a single bullet, you've already got a full clip ready to shoot.

In this video, around 2 minutes in, you can see a discussion and then a demonstration of the Lebel Model 1886, which the French used throughout the war. While it is a bolt action, it loads just like a lever action. When you compare it to the bolt action, magazine-fed German rifles the French army was going up against, there's no competition. You can see some of those rifles demonstrated in this video. Also, it should be noted that bolt action rifles provide greater accuracy than the lever action because of how the bolt holds the bullet in place.

Finally, I'd like to speak simply as someone who has handled these types of weapons personally. When it comes to durability and reliability, the bolt action is far superior. I've had misfire issues using clean and well-maintained lever actions that spent most of their time inside on gun racks - I can't imagine how badly they would perform on a battlefield, especially one like the Western trenches of WWI. Keeping your gun free of dirt would be an impossible task, as the mud and filth would get everywhere, and once inside the tube magazine of a Lebel, it couldn't be easily cleaned out. Comparatively, the Americans' M1917 Enfield won't clog nearly as easily and cleans out much faster.

With all of these advantages, it's easy to see why the US used bolt action rifles when they entered the war.

Building on the remarks of /u/jamesjamersonson, bolt action rifles are also significantly easier to fire from the prone position. Due to the mechanism of bolt action rifles being located on top of the rifle, rounds can be cycled without shifting the aim of the rifle dramatically. To cycle a lever action, the rifle would either need to be turned to the side or pointed upwards, both would mean more time would be required to reacquire a target. Warfare on the Western Front would often entail defending trenches, this in turn would necessitate resting the rifle on the parapet in order to steady the rifleman's aim. A lever action would only be a hindrance in this situation.