diff --git a/compiler/src/dotty/tools/dotc/ast/tpd.scala b/compiler/src/dotty/tools/dotc/ast/tpd.scala
index 514377802e57..10f3fe634495 100644
--- a/compiler/src/dotty/tools/dotc/ast/tpd.scala
+++ b/compiler/src/dotty/tools/dotc/ast/tpd.scala
@@ -889,9 +889,15 @@ object tpd extends Trees.Instance[Type] with TypedTreeInfo {
       else Erasure.Boxing.adaptToType(tree, tp)
 
     /** `tree ne null` (might need a cast to be type correct) */
-    def testNotNull(implicit ctx: Context): Tree =
-      tree.ensureConforms(defn.ObjectType)
-        .select(defn.Object_ne).appliedTo(Literal(Constant(null)))
+    def testNotNull(implicit ctx: Context): Tree = {
+      val receiver = if (defn.isBottomType(tree.tpe)) {
+        // If the receiver is of type `Nothing` or `Null`, add an ascription so that the selection
+        // succeeds: e.g. `null.ne(null)` doesn't type, but `(null: AnyRef).ne(null)` does.
+        Typed(tree, TypeTree(defn.AnyRefType))
+      }
+      else tree.ensureConforms(defn.ObjectType)
+      receiver.select(defn.Object_ne).appliedTo(Literal(Constant(null)))
+    }
 
     /** If inititializer tree is `_', the default value of its type,
      *  otherwise the tree itself.
diff --git a/compiler/src/dotty/tools/dotc/transform/PatternMatcher.scala b/compiler/src/dotty/tools/dotc/transform/PatternMatcher.scala
index 65aadb86ccb9..00ebc03cff46 100644
--- a/compiler/src/dotty/tools/dotc/transform/PatternMatcher.scala
+++ b/compiler/src/dotty/tools/dotc/transform/PatternMatcher.scala
@@ -335,10 +335,16 @@ object PatternMatcher {
               patternPlan(casted, pat, onSuccess)
             })
         case UnApply(extractor, implicits, args) =>
-          val mt @ MethodType(_) = extractor.tpe.widen
-          var unapp = extractor.appliedTo(ref(scrutinee).ensureConforms(mt.paramInfos.head))
-          if (implicits.nonEmpty) unapp = unapp.appliedToArgs(implicits)
-          val unappPlan = unapplyPlan(unapp, args)
+          val unappPlan = if (defn.isBottomType(scrutinee.info)) {
+            // Generate a throwaway but type-correct plan.
+            // This plan will never execute because it'll be guarded by a `NonNullTest`.
+            ResultPlan(tpd.Throw(tpd.Literal(Constant(null))))
+          } else {
+            val mt @ MethodType(_) = extractor.tpe.widen
+            var unapp = extractor.appliedTo(ref(scrutinee).ensureConforms(mt.paramInfos.head))
+            if (implicits.nonEmpty) unapp = unapp.appliedToArgs(implicits)
+            unapplyPlan(unapp, args)
+          }
           if (scrutinee.info.isNotNull || nonNull(scrutinee)) unappPlan
           else TestPlan(NonNullTest, scrutinee, tree.span, unappPlan)
         case Bind(name, body) =>
diff --git a/tests/run/i5067.check b/tests/run/i5067.check
new file mode 100644
index 000000000000..7f5dbdea750f
--- /dev/null
+++ b/tests/run/i5067.check
@@ -0,0 +1,6 @@
+matches null literal
+matches null literal
+not implemented
+match error
+not implemented
+match error
diff --git a/tests/run/i5067.scala b/tests/run/i5067.scala
new file mode 100644
index 000000000000..663ccb15e39a
--- /dev/null
+++ b/tests/run/i5067.scala
@@ -0,0 +1,47 @@
+// Test that we correctly handle scrutinees with type `Null` or `Nothing`.
+object Test {
+  def main(args: Array[String]): Unit = {
+    null match {
+      case Some(_) => println("matches Some")
+      case (_, _) => println("matches Pair")
+      case null => println("matches null literal")
+    }
+
+    type X = Null
+    (null: X) match {
+      case Some(_) => println("matches Some")
+      case (_, _) => println("matches Pair")
+      case null => println("matches null literal")
+    }
+
+    type Y = Nothing
+    try {
+      (??? : Y) match {
+        case Some(_) => println("matches Some")
+        case _ => println("matches anything")        
+      }
+    } catch {
+      case e: NotImplementedError => println("not implemented")
+    }
+
+
+    try {
+      val Some(_) = null
+    } catch {
+      case e: MatchError => println("match error")
+    }
+
+    try {
+      val Some(_) = ???
+    } catch {
+      case e: NotImplementedError => println("not implemented")
+    }
+    
+    val x: X = null
+    try {
+      val Some(_) = x
+    } catch {
+      case e: MatchError => println("match error")
+    }
+  }
+}
diff --git a/tests/run/i5067b.check b/tests/run/i5067b.check
new file mode 100644
index 000000000000..be60186f7c25
--- /dev/null
+++ b/tests/run/i5067b.check
@@ -0,0 +1,3 @@
+match error
+match error nested
+not implemented error
diff --git a/tests/run/i5067b.scala b/tests/run/i5067b.scala
new file mode 100644
index 000000000000..cd1f3898cb7e
--- /dev/null
+++ b/tests/run/i5067b.scala
@@ -0,0 +1,34 @@
+// Test that we correctly handle scrutinees with type `Null` or `Nothing`.
+object Test {
+  def main(args: Array[String]): Unit = {
+      class B[T] {}
+      object B {
+        def unapply[T](x: Any): Option[B[T]] = None
+      }
+      try {
+        val B(_) = null
+      } catch {
+        case e: MatchError => println("match error")
+      }
+
+      null match {
+        case null =>
+          try {
+            null match {
+              case Some(_) => ()
+            }
+          } catch {
+            case e: MatchError => println("match error nested")
+          }
+      }
+
+      try {
+        ??? match {
+          case (_, _) => ()
+          case _ => ()
+        }
+      } catch {
+        case e: NotImplementedError => println("not implemented error")
+      }
+  }
+}