Yes they can, a trait that extends a class puts a restriction on what classes can extend that trait - namely, all classes that mix-in that trait must extend that class.
scala> class Foo
defined class Foo
scala> trait FooTrait extends Foo
defined trait FooTrait
scala> val good = new Foo with FooTrait
good: Foo with FooTrait = $anon$1@773d3f62
scala> class Bar
defined class Bar
scala> val bad = new Bar with FooTrait
<console>:10: error: illegal inheritance; superclass Bar
is not a subclass of the superclass Foo
of the mixin trait FooTrait
val bad = new Bar with FooTrait
^