Light Yagami was on a roll in the first few episodes of Death Note. Within the first few days of obtaining the Death Note, he had written down hundreds of names. He then caught the eye of the world governments and was proclaimed enemy number one of super detective L. Not bad for someone still in high school.
In Death Note season 1, Episode 5, "Tactics," FBI agent Raye Penber is sent to investigate the Kira case and ends up suspecting that Light Yagami is Kira. However, his doubts are assuaged when he notices that Light seems no different from any other high school student after tailing him and boarding a bus while Light is on a date. Things take a turn for the worse when the bus that Raye and Light are on is highjacked -- which was all part of Light's plan to steal Raye's identity. This, of course, fares poorly for Raye, and if he had avoided one small blunder, he might have been the detective who cracked the Kira case.
The one small blunder Raye made was that he didn't listen to his fiancee -- if he had, he might've actually lived. Raye's fiancee was none other than Naomi Misora, a brilliant former FBI agent who saw something fishy about what happened during the bus highjacking after noticing too many odd coincidences. Naomi is shown throughout the series as having excellent deduction skills and acute intuition, but Raye quickly shot her down and told her to no longer worry about the Kira case, saying that he had a handle on it -- and that was his downfall.
Unfortunately, Naomi took her soon-to-be husband's words to heart and no longer pursued the case. All of that changed when, a few days later, Raye and all of the twelve FBI agents assigned to the Kira case were killed by Light. A heartbroken Naomi decides to crack the case herself, and Light, catching wind of this, kills her as well after a tense game of cat and mouse, proving that she was more than capable of catching him.
All of this tragedy could have been avoided if Raye had sat down and did not let his worries over Naomi's safety get the better of him. There was no way that Light could know that the two of them would go to the Kira investigation team in private and disclose Naomi's suspicions of what happened on the bus that day. Because Naomi also worked with L in the past on a previous case, L would have been more than happy to hear her out and would have been able to deduce earlier that Light is indeed Kira.
Raye was worried for Naomi's life and his devotion to and wanting a better life for her was an admirable act of love. But Naomi was a detective long before she met Raye, and hearing out her suspicion would have not only saved his life, but also Naomi's and the lives of his fellow FBI agents. If Raye had done that, Kira would have been caught much sooner. But then again, that would also mean that an excellent series like Death Note would have been cut short.