Checking if a text contains N consecutive repeating characters
Edit - there is a bug with this: see my other answer

This is very easy with Regular Expressions:

public static bool HasConsecutiveChars(string source, int sequenceLength)
{
    // should check sequence length

    // just repeating letters
    return Regex.IsMatch(source,"([a-zA-Z])\\1{"+ (sequenceLength - 1) + "}");

    // any character version
    // return Regex.IsMatch(source,"(.)\\1{"+ (sequenceLength - 1) + "}");
}


The regular expression simply captures each letter into a group and then checks if it is repeated the number of times minus one. I've omitted range checking.

Talking about your code for a moment:

Great method name - would be perfect as an extension method in my opinion.

These two checks can be merged: 

if (string.IsNullOrEmpty(source))
{
    return false;
}

if (source.Length == 1)
{
    return false;
}