In general, that should be fine for most larger carriers. However, it does depend on your carrier. The best thing to do is to check on any of the airlines you might be flying to determine what their specific requirements are, as they can vary significantly across carriers.

As mentioned, Easyjet (and some other European low-cost carriers) will only allow one bag, period, and that bag must meet certain size/weight requirements.

In the US, Spirit is the most strict, allowing a single personal item (size restrictions) for free, and a true carryon being an additional charge.