Asalamu wa alaikum,
Here is the Ayah you gave in your question:
And do not insult one another and do not call each other by [offensive] nicknames. Wretched is the name of disobedience after [one's] faith. Qur'an 49:11
Even though the Ayah has nothing to do with bad nicknames, the answer the Ayah, Wretched is the name of disobedience after [one's] faith. It is not good to have bad nicknames or even that names, because as ashes999 keeps saying they affect your psychology, and trust me they realy do.
يَـٰٓأَيُّہَا ٱلَّذِينَ ءَامَنُواْ لَا يَسۡخَرۡ قَوۡمٌ۬ مِّن قَوۡمٍ عَسَىٰٓ أَن يَكُونُواْ خَيۡرً۬ا مِّنۡہُمۡ وَلَا نِسَآءٌ۬ مِّن نِّسَآءٍ عَسَىٰٓ أَن يَكُنَّ خَيۡرً۬ا مِّنۡہُنَّۖ وَلَا تَلۡمِزُوٓاْ أَنفُسَكُمۡ وَلَا تَنَابَزُواْ بِٱلۡأَلۡقَـٰبِۖ بِئۡسَ ٱلِٱسۡمُ ٱلۡفُسُوقُ بَعۡدَ ٱلۡإِيمَـٰنِۚ وَمَن لَّمۡ يَتُبۡ فَأُوْلَـٰٓٮِٕكَ هُمُ ٱلظَّـٰلِمُونَ (١١)
O ye who believe! let not some men among you laugh at others: it may be that the (latter) are better than the (former): nor let some women laugh at others: it may be that the (latter) are better than the (former): nor defame nor be sarcastic to each other, nor call each other by (offensive) nicknames: Ill-seeming is a name connoting wickedness, (to be used of one) after he has believed: And those who do not desist are (Indeed) doing wrong. (11)
Narrated AbuJubayrah ibn ad-Dahhak:
This verse was revealed about us, the Banu Salimah: "Nor call each
other by (offensive) nicknames: ill-seeming is a name connoting
wickedness (to be used of one) after he has believed." He said: When
the apostle of Allah () came to us, every one of us had two or three
names. The Apostle of Allah () began to say: O so and so! But they
would say: Keep silence, Apostle of Allah! He becomes angry by this
name. So this verse was revealed: "Nor call each other by (offensive)
nicknames."
حَدَّثَنَا مُوسَى بْنُ إِسْمَاعِيلَ، حَدَّثَنَا وُهَيْبٌ، عَنْ
دَاوُدَ، عَنْ عَامِرٍ، قَالَ حَدَّثَنِي أَبُو جُبَيْرَةَ بْنُ
الضَّحَّاكِ، قَالَ فِينَا نَزَلَتْ هَذِهِ الآيَةُ فِي بَنِي سَلِمَةَ
{ وَلاَ تَنَابَزُوا بِالأَلْقَابِ بِئْسَ الاِسْمُ الْفُسُوقُ بَعْدَ
الإِيمَانِ } قَالَ قَدِمَ عَلَيْنَا رَسُولُ اللَّهِ صلى الله عليه
وسلم وَلَيْسَ مِنَّا رَجُلٌ إِلاَّ وَلَهُ اسْمَانِ أَوْ ثَلاَثَةٌ
فَجَعَلَ النَّبِيُّ صلى الله عليه وسلم يَقُولُ " يَا فُلاَنُ " .
فَيَقُولُونَ مَهْ يَا رَسُولَ اللَّهِ إِنَّهُ يَغْضَبُ مِنْ هَذَا
الاِسْمِ فَأُنْزِلَتْ هَذِهِ الآيَةُ { وَلاَ تَنَابَزُوا
بِالأَلْقَابِ }
.