Can One Be A Software Engineer And Web Developer The Same Time?

Uhmmm,

Yes you can  

Let us get one thing straight though - there is a difference b/w a web designer (css/html/javascript, etc) and a web developer (css/html, javascript/xml/php/jsp, etc) 

If you are building web applications and not merely websites, then you are a bona fide web developer (you combine your knowledge of the back end with the front end).  But if all you do for a living is build basically what amounts to informational sites (no database interactions involved - "click here" to go "there", with a few "wow" effects thanks to FLASH, JAVAscript and perhaps AJAX), then you are a web designer.

A web developer is sort of, to me, a semi-software engineer, in that the person acts as the "middle man" between the actual back end guys and the front end.  They know for example, enough JAVA to access the business logic, exposed via JAVA tags (ala JSP, again, for example, since this is what I do), as well as enough of the front end to display the returned data in an elegant and efficient manner.

Left for me though, I would rather have the software engr and web developer roles separate.  Though it is good to be a jack of all trades, if I were an employer, I would rather have experts in the 2 fields.  That way, with less work, they have the time to specialize in their respective fields.   In a place like the states, where they want you to put in more hours doing more work for the same pay, I would rather work smart than hard, personally, for what pretty much works out to be the same pay.  So though I can say I know enough of both worlds, I rather stay on the "easy (not so easy anymore thanks to AJAX, AFLAX, etc - it seems once you know one thing, three more things pop up  ).  Having said that, there is nothing wrong in being both.

I hope this helps.