The European diplomatic system was arguably falling out of Bismarck's control by 1887, a state of affairs best exemplified by the Reinsurance Treaty of that year: while it successfully reaffirmed Russian neutrality were Germany to be attacked by France, the price of it was a secret clause it which Germany agreed to remain neutral in the event of a Russian move on the Dardanelles. Britain would never accept this, and it contradicted the earlier Mediterranean Agreements that Germany had signed to preserve the status quo in the Mediterranean.
To understand why, we need to know a bit about Bismarck's philosophy of foreign policy. The Kissingen Memorandum is the closest thing to a political testimony from Bismarck, and details his plans to freeze France out of Europe and prevent a coalition (possibly Britain, France and Austria, but worst of all Austria, Russia and France) forming against Germany. Among the methods he described in the Memorandum was negotiating a suitable status quo in the eastern Mediterranean and the Middle East between Britain and Russia (i.e.: preserving British naval supremacy and the route to India while facilitating Russia's Balkan policy, in particular their search for a warm-water port), and seeking to divide Britain from France over their differences over Mediterranean policy. Until 1887 this Mediterranean policy largely succeeded: the Mediterranean Agreements reaffirmed the British position on Gibraltar, Egypt and Cyprus, de facto halted French colonial expansion along the North African coast, and had the potential to bring in British naval support if Germany's Triple Alliance ally Italy was attacked by France. The British were further sweetened by the Agreements further reaffirming the neutrality of the Dardanelles.
That same year, however, the Three Emperor's League of Germany, Russia and Austria split over differences between Russian and Austrian policy in the Balkans (which both saw as being within their sphere of influence as the Ottoman Empire declined). Bismarck managed to reaffirm Russian neutrality with the Reinsurance Treaty, but the prospect of Russian troops in Constantinople controlling the straits from the Black Sea to the Mediterranean, bearing down on the Suez Canal and the overland route to India, was one the British would never have accepted. Bismarck stored up a great contradiction in his system that would only really come into play after his dismissal: in 1890 the treaty was up for renewal, but Kaiser Wilhelm II let the treaty lapse in the hopes of improving relations with Britain and believed that his personal relationship with the Tsar would be enough to keep the peace. He was mistaken, and two years later, Russia concluded a military alliance with France, overturning all of Bismarck's foreign policy achievements. It is the Iron Chancellor's fortune that he was not in office to take the responsibility that really was his.