Question: Could I say: 

* a thousand ten (1010),
* a thousand a hundred eleven(1111)

or in these cases I must use **one thousand** instead of **a thousand**? Would using "a" be grammatically incorrect?

Answer: 
Br Eng would be 

> 
> One thousand and ten  
> 
>  One thousand, one hundred and eleven
> 
> 
> 

US Eng seems to miss out the 'and' a lot of the time, but I've never figured the exact pattern, or if it's dialectical, but I've heard such as

> 
> One thousand ten  
> 
>  One thousand, one hundred eleven
> 
> 
> 

but I wouldn't use my guess at how the US does it, I'm a Brit ;)

Neither, though, would use 'a' in that situation, they'd use 'one'.

*From comments:*  

If we are using "whole" numbers, such as 100, 1,000, 1,000,000 etc, then 'a' could be used. A thousand or a million is perfectly acceptable. It just tends not to be used when the number is more 'complex'.
"a thousand a hundred and ten" is *grammatical* but not *idiomatic* in written English. A native speaker would be likely to spend a moment confused, but would understand what you meant.

It is, however, idiomatic in some *spoken* dialects.

If you're going to use "a", use it *consistently*. Don't use "one" for some parts of the number and "a" for others.