Otto von Bismarck, the principal statesman behind the unification of Germany, and the architect of a political balance that led to relative peace in Europe for 40 years, saw it coming from very far away indeed. In 1897, a year before his death, he predicted both a great defeat for Germany 20 years after he died and that the "great European war" would "come out of some damned foolish thing in the Balkans".

In fact, he explicitly warned the new emperor who had forced him to resign, Wilhelm II, that deriving government policy based on the intrigues of his diplomats and officers would be dangerous, and that it would only work as long as he still had the officers cultivated by Bismarck. This was extremely prescient; during the July Crisis in 1914, every politician and officer involved knew that a great war was imminent if the situation was not defused. If Wilhelm II had maintained a cohesive policy and listened to his own instincts, the Great War could have been averted, at least for the time being; he wanted Austria-Hungary to stand down after Serbia complied with the ultimatum provided to it, as further aggression could not possibly be justified by this point. However, because of his militarist officers and diplomat corps, especially one Helmuth von Moltke, he was ultimately coerced into signing the order to mobilize, which was as good as declaring war.

By all accounts, von Bismarck was a political genius, and you asked about 1913, not 1897. What about people who weren't von Bismarck? Well, as I touched on in when discussing the July Crisis in the context of von Bismarck's predictions, the statesmen involved knew what was going to happen. The German militarists, in particular, very specifically wanted a world war. Both France and Russia were not prepared for war, especially Russia which was in the midst of a massive modernization program, and they expected that by 1917 they would no longer be able to assert their dominance over Russia, but that they would win if they were able to start a justified war before then. The Austrians did not necessarily want a world war, but were notoriously indecisive and the German "blank cheque" for support effectively encouraged them to push the limits with Serbia, as they felt a great victory was needed to hold together their country in the midst of its decline. It was especially feared that if Austria continued to show indecisiveness that Germany would not continue to support her.

The Russian, French, and British diplomats all knew that war in Serbia would lead to a world war, and tried very hard to defuse the situation throughout the month of July. The French president and prime minister held a summit with Russia, and many parties worked frantically to try to find a solution. British diplomats, in particular, were deceived into believing that German diplomats were genuinely negotiating for peace as well, and that peace might yet be achieved. Russian diplomats suspected otherwise, and rightfully so, as although German diplomats tried to put on appearances of working to resolve the Austro-Serbian crisis, they were exploiting it to start war. Nicholas II, last tsar of Russia and third cousin of Wilhelm, desperately tried to convince Wilhelm on a personal level to avert war, but ultimately Wilhelm's officers pushed him into it. Serbia went so far as to accept the humiliating ultimatum that was intentionally crafted to be rejected, but this was in vain as Austria and Germany were at this point committed to causing war.

The July Crisis was when things came to a boil, but the situation was simply the spark to light the kindling, which people were fully aware of. The aforementioned German officers and diplomats had been planning and itching to make the war happen since at least 1912. Austria-Hungary had been planning for a war with Serbia since 1912 as well, long before the assassination of Archduke Ferdinand. You could say that they were not only aware of the kindling, but actively preparing for a spark that would not make them the obvious arsonists. The triple Entente knew of the potential for war and the scope of it, but the general thinking was that peace could be mediated to prevent it from actually happening.

When war did break out, the gravity of what was about to unfold was fully understood; for example British diplomat Edward Grey remarked days after it began that "the lamps are going out all over Europe, we shall not see them lit again in our life-time". Not only among statesmen but also the populace; author H.G. Wells described it first in newspaper articles and then a book as "the war to end all wars", within the first month of the war, and this phrase was widespread throughout the public consciousness.

In short, people saw it coming and tried to prevent it to the best of their abilities, but it's difficult to prevent a powderkeg from exploding when one party is deliberately, discreetly trying to light it because they think the explosion will work to their benefit.

Sources:
Germany's Aims in the First World War Fritz Fischer 1967
Europe's Last Summer: Who Started the Great War in 1914? David Fromkin 2004

Certainly. Many people saw WWI coming, since it was obvious that the arms races that were happening (Dreadnoughts, artillery, machine guns, airplanes, etc.) combined with the growth of expansive mutual defense alliances between great powers weighed against those powers being brought into increasing conflict with one another due to scrambles for empire made it fairly obvious that some sort of war was inevitable.

Moreover, it's important to keep in mind that prior to WWI war between European powers was common, normal, and routine and thus expected. It was expected that great powers would come into conflict, it was expected that warfare would be used to resolve said conflict, it was expected that the resolution of the warfare would involve changes in relative power between great powers, and so forth. The 19th century was replete with European conflicts, as was the 18th, 17th, 16th, 15th, 14th, etc. Just highlighting the main conflicts of the century prior to WWI you have the Napoleonic Wars, you have the Crimean War, you have the Franco-Prussian War, you have the various conflicts around the rise of Prussia and the unification of Germany as well as the unification of Italy. And those are just the extreme highlights. If you dig into the period there's practically a fractal pattern of conflicts that show up. What was unanticipated about The Great War (though obvious to anyone who really looked at the matter closely) was the degree to which it was all encompassing and the vast amount of destruction it caused. But that's what happens when you bring fully industrialized world class economies in conflict with one another and you flood the battlefield with millions of soldiers fighting with artillery and machine guns (millions die, normal life grinds to a halt, millions more die from starvation and disease, etc.)

As for WWII, people saw it coming as well, but this time they had the recent trauma of WWI and so were deathly afraid of a similar conflict erupting. They tried to avoid it or convince themselves it wasn't inevitable. You see a lot of coverage of Hitler, for example, trying to find a way to imagine his ascent to power as less alarming than it actually was, trying to pick and choose examples of his actions that indicated he was somewhat of a reasonable or civilized individual, able to listen to reason, unlikely to act overly rashly or violently, etc. Additionally, during the 1920s especially there were many attempts to forestall the sorts of arms races and international conflicts that had led to WWI. There was the development of the Washington Naval Treaty, keeping a lid on naval development. There was the League of Nations, providing a venue for working out conflicts through diplomacy rather than war, and so forth. And many of the actions in the run up to WWII were done due to a fear of igniting a new Great War. When Germany remilitarized the Rhineland the French and British did not respond militarily, because they didn't want to start a war, they hoped instead that the actions were the end of something rather than the beginning. When Japan used a false flag attack on itself to justify an invasion of China the truth was discovered and openly reported, but nothing of substance was done about it through the League of Nations. In the end none of the temporizing or exercises of diplomacy and soft-power (such as the American boycott of Japan) could put the brakes on the world conflict.

However, in both cases exceptionally few people seriously foresaw the wars that were to come. In the case of WWI even though many people saw the conflict as inevitable, many people were welcoming of the conflict, thinking it would be short and beneficial to "their side". In contrast, a great many people saw WWII coming, and saw that it would be as disastrous as The Great War, but precisely because of that sentiment, which they felt every civilized individual felt as well, they thought that no one would be crazy enough to plunge into such a calamitous conflict and would strive to avoid it, thus thinking the actuality of another great war to be unlikely.

Considering one specific event, two US Navy Admirals did foresee the successful Japanese attack on Pearl Harbor. Japan had long been considered a serious potential rival in the Pacific and detailed plans for war against them had been in the work for decades. (Plan Orange)

In 1932, Admiral Harry E. Yarnell proved in war games that Pearl Harbor was vulnerable to Japanese carrier-born attack - at a time when few even accepted that a plane could sink a major ship. But his warnings were dismissed. In 1938, Admiral Ernest J. King confirmed Yarnell's findings, which were again dismissed.

The attack three years later was no surprise for these two, of course. At the time Yarnell had retired, while King was Commander in Chief of the Atlantic Fleet. Roosevelt soon raised King to Commander in Chief of the Navy and Chief of Naval Operations, and Yarnell was brought out of retirement and eventually served on King's staff in 1943.