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 ^