Yes, that is just how I started. I went off wheat because it seemed to give me indgestion (and worse!). I started to lose weight. So 11 months ago I started lowering the carbs generally. I read the Atkins book, to check out the theory and get more confident and then just did it.

In the 60s, before the low fat thing took over, my Dad lost weight with the Yudkin diet. Dr Yudkin was an early campaigner against sugar (he wrote a book called 'Pure, white and deadly') who devised a diet where he gave food carbohydrate units. These were 5g. So an apple was 2, and a slice of bread 3 etc. He suggested starting with 15 and seeing what happened. if you lost, good, if not, try 12 etc.
Advantages; simple to count and it put the burden on you to watch the results. If you weren't losing, cut the carbs. He said this got round people's tendency to 'forget'what they had actually eaten, or miss the thickening in soups etc.
Disavantages; None, really. You need to know your food values, but that applies to all ways of eating. Atkins stiff way of starting give people success which always helps. Yudkin may be slower to get going.

I really value Atkins and this site, for reassurance. I think the need to drink lots is the greatest discovery for me from the Atkins ideas.