If logcat is capping the length at 1000 then you can split the string you want to log with String.subString() and log it in pieces. For example:

int maxLogSize = 1000;
for(int i = 0; i <= veryLongString.length() / maxLogSize; i++) {
    int start = i * maxLogSize;
    int end = (i+1) * maxLogSize;
    end = end > veryLongString.length() ? veryLongString.length() : end;
    Log.v(TAG, veryLongString.substring(start, end));
}