Yes, a symbol can have both own-values and down-values, but it is usually bad practice to give a symbol both. Further, the order in which they are created matters, but there are problems both ways.

###OwnValue first

 y = 42;
 OwnValues @ y
>`{HoldPattern[y] :> 42}`

But now you can't set a own-value, because 

 y[42] = 0
>Set::write: Tag Integer in 42[42] is Protected. 
>`0`

 DownValues @ y
>`{}`

###DowValue first

 x[42] = 0;
 DownValues @ x

>`{HoldPattern[x[42]] :> 0}`

 x = 42;
 OwnValues @ x
>`{HoldPattern[x] :> 42}`

The symbol `x` has both an own-value and a down-value, but look what happens when `x[42]` is evaluated.

 x[42]
>`42[42]`

This happens because under normal evaluation rules, Mathematica evaluates the head `x` of `x[42]` before anything else. When evaluating a symbol such as `x`, it looks at own-vales first and finds one. It uses that own-value to replace `x` with 42. It next evaluates the parts (arguments) of expression and of course gets 42, and so `x[42]` evaluates to `42[42]`. Mathematica doesn't look for a down-value of `x` because it doesn't need to.