Letting the nails grow long is makrooh, if not haraam, because the Prophet (peace and blessings of Allaah be upon him) set a time limit for cutting the nails, and said that they should not be left for more than forty days. (Narrated by Muslim in al-Tahaarah, 258).
It is strange that those who claim to be civilized leave their nails to grow even though they trap dirt and make a person look like an animal. The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Use whatever causes blood to flow, and eat the animals if the name of Allaah is mentioned on slaughtering them. Do not slaughter with teeth or fingernails and I will tell you why: it is because teeth are bones (i.e., cannot cut properly) and fingernails are the tools used by the Ethiopians (whom we should not imitate for they are infidels).” (Narrated by al-Bukhaari in al-Sharikah, 3507; Muslim in al-Adaahi, 1968). This refers to the fact that they used to use their fingernails as knives to slaughter animals and cut the meat, etc. This is the way of those who resemble animals.