Sometimes but it would be rare for either side to dig a complete trench closer to an enemy trench. Such a practice was incredibly dangerous and only brought either side closer to the other's artillery. What did happen was prior to large attacks, trenches would be driven forward perpendicular to the existing trench and would be used by the attacking side to get as close as possible to the enemy trench before they had to stick their heads above ground level.

Now, as to how either side gained ground during the First World War, it wasn't really until 1916 that either side was really able to after trench warfare set in. The presence of artillery and machine guns allowed the defending side (usually the Germans) to prevent any successful attacks on their own trenches. This wasn't always the case, there were cases where the line advanced a bit during 1915 but for the most part, it was pretty stagnant with neither side capable of shifting the other. This changed in 1916 when larger quantities of artillery, specifically howitzers and large guns became available. When the belligerent nations went to war, most nations relied on fields guns to provide artillery fire. These guns, such as the British 18-pounder, the French 75mm MLE and the German 7.7 cm FK 96 n.A., were superb fields guns, capable of rapid rates of fire and were highly mobile but they fired their shells at a very low trajectory. This made them perfect weapons for use against enemy infantry but they were not suited for use against trenches and other earthworks. For that, a weapon was needed that would fire a high explosive shell high into the air, travel along a parabolic flight path and then drop almost vertically into an enemy trench. Howitzers were capable of doing this. The problem was that howitzers had largely fallen out of favour amongst continental armies and those that were available were of relatively light caliber. By 1916, larger howitzers and field guns were arriving in large numbers on the Western Front.

The arrival of these weapons opened up new possibilities. Army commanders now had the potential to destroy enemy positions and infantry within their own trenches, allowing their own infantry to advance relatively unmolested towards the enemy trench. That was the theory anyway. The first day of the Somme showed that more needed to be done to destroy enemy positions and their occupants then simply firing millions of shells across No Man's Land. Barbed wire had to be more adequately cut and heavier guns and howitzers needed to be allocated to the destruction of enemy trenches. Additionally, the destruction or neutralisation of the enemy artillery also had to accomplished or advancing infantry would be cut down by shrapnel fired by enemy guns. The Somme taught the British some very costly lessons about how best to use their artillery and thankfully they learned from those lessons. By 1917, the British were utilising their artillery in a way that allowed them to achieve some stunning victories. More resources were spent on detecting and locating enemy artillery before a major attack, more howitzers were allocated to the destruction of enemy trenches and more field guns were employed to cut the enemy barbed wire. Added to this was the refinement of the Creeping Barrage, an artillery tactic wherein friendly artillery would lay down a barrage just ahead of advancing infantry. That barrage would move forwards according to a precise timetable with the infantry following behind. As the barrage passed over the enemy defences, the defenders would be forced to take cover. By the time the barrage had passed and the defenders were emerging from the cover, the attacking infantry would be upon them. This prevented the defenders from utilising the deadly machine guns. By 1917, the Creeping Barrage was a highly refined and scientific concept and was being used with amazing success, not least of which was the Canadian and British victory at Vimy Ridge.

Now, this didn't mean that attacks were made easy. The German army didn't rest on their laurels and let themselves be bombed into oblivion. Instead, they changed their defensive doctrine. Instead of focusing their manpower in the frontline trenches in range of the enemy artillery, they shifted to a doctrine known as Defence in Depth. This doctrine saw the Germans only position a small number of men in their frontline trenches with the bulk of their forces positioned further to the rear, out of range of the enemy artillery. When the Allied armies attacked, they could easily overcome the frontline defences using their new artillery techniques but soon found that in order to reach the next German positions, they had to advance over open ground covered in barbed wire and in range of the enemy artillery. By the time they had fought through this, taking casualties from artillery and machine guns along the way, they would come up against the much more heavily defended German positions and to make matter worse, they were out of range of their own artillery so there was no Creeping Barrage to help them. It was at this point that the German forces would counterattack the weakened Allied infantry, driving them back across the ground they had captured. This was a highly successful strategy for the German Army and it would not be until 1918 that the Allies would finally be able to breach these new defences and still only at an enormous cost of life to the advancing infantry.

Sources:

Firepower by Dominick Graham and Shelford Bidwell

Artillery in the Great War by Paul Strong and Sanders Marble

Battle Tactics of the Western Front - The British Army's Art of Attack 1916-18 by Paddy Griffith

The Killing Ground by Tim Travers