If you think about the security dilemma as something that causes conflict in general (as opposed to outright war), it is easier to think of examples from the last 20 years. There are all sorts of conflicts (that may some day cause war) that exist today due in part to actions taken by certain countries to improve their security.
For example, think of the tensions over nuclear programs in North Korea and Iran or, for that matter, over Saddam Hussein's WMD programs in Iraq. In each case, the country undertaking the programs seeks to increase its security. In each case, it has come into conflict with the United States since the US sees these countries' possession of WMD as a threat to its interests.
You can also think of China and the US in this context. For example, China has just launched an aircraft carrier, its first. China is, one can argue, simply trying to increase its security by acquiring the capability of projecting force. However, this acquisition has definitely increased tensions with the US as the US starts to worry about whether China will become a threat to US interests in the Pacific.