It wasn't the Russians who dropped the relationship (it wasn't an alliance) with Germany, it was the Germans. More specifically, it was Kaiser Wilhelm II. And it didn't happen in 1902 but in 1890.
In 1887, Bismarck negotiated a diplomatic masterpiece, a secret treaty with Russia against his ally Austria. Called the Reinsurance Treaty, it was defensive and promised only neutrality, not military assistance, if either party were attacked (German neutrality if Austria attacked Russia, Russian neutrality if France attacked Germany). Despite its limitations, it violated, as Bismarck well knew, the alliance treaty Germany had with Austria. So Bismarck obviously insisted on secrecy. Tsar Alexander III was no less anxious to hide the existence of the Reinsurance Treaty. Himself a pan-Slav, he could predict the reaction of other pan-Slavs. Alexander signed the treaty only because it gave him a promise of German neutrality in case Austria provoked a war with Russia. Russia did not want to fight Germany; certainly the Russian Army was not capable of fighting Germany and Austria together.
In 1890, the Treaty was up for renewal. Wilhelm had just deposed Bismarck as Chancellor (the reasons why are long and involved but basically amount to Wilhelm's dislike of anyone being as powerful as he thought he was). Wilhelm had promised the Russian ambassador, Shuvalov, that the treaty would be renewed. The new German Chancellor, Georg von Caprivi (don't ask about the Italian name), was informed about the Treaty by the eminance gris of the Germany Foreign Ministry, Freidrich von Holstein. Holstein had hated Bismarck (Holstein hated almost everyone) and warned Caprivi that the Treaty undermined the Austrian alliance. Caprivi talked to Wilhelm who, ignoring what he told Shuvalov just two days previously, told Caprivi to let the Treaty lapse.
Stripped of the relationship with Germany, Russia began to seek another. On July 23, 1890, only four months after Germany's refusal to renew the Reinsurance Treaty, a French naval squadron made a visit to the Russian naval base at Kronstadt. Tsar Alexander gave a dinner to the French officers and stood during the playing of the Marseillaise, the revolutionary anthem which had been prohibited within the borders of the Russian Empire. Thus began the Franco-Russian Double Entente.
