diff --git a/core-blocks/heading/edit.js b/core-blocks/heading/edit.js
index 60869aca35e3f1..a682629abaf614 100644
--- a/core-blocks/heading/edit.js
+++ b/core-blocks/heading/edit.js
@@ -15,6 +15,7 @@ import './editor.scss';
 
 export default function HeadingEdit( {
 	attributes,
+	canInsertSibling,
 	setAttributes,
 	mergeBlocks,
 	insertBlocksAfter,
@@ -66,6 +67,13 @@ export default function HeadingEdit( {
 				onSplit={
 					insertBlocksAfter ?
 						( before, after, ...blocks ) => {
+							if ( ! canInsertSibling( 'core/paragraph' ) ) {
+								// If it is not possible to insert a paragraph
+								// don't split the block just try to insert pasted blocks after it.
+								// Unsupported blocks are ignored.
+								insertBlocksAfter( blocks );
+								return;
+							}
 							setAttributes( { content: before } );
 							insertBlocksAfter( [
 								...blocks,
diff --git a/core-blocks/list/index.js b/core-blocks/list/index.js
index 8c3924f87a15e8..02ae5a582a36fa 100644
--- a/core-blocks/list/index.js
+++ b/core-blocks/list/index.js
@@ -252,6 +252,7 @@ export const settings = {
 		render() {
 			const {
 				attributes,
+				canInsertSibling,
 				insertBlocksAfter,
 				setAttributes,
 				mergeBlocks,
@@ -302,6 +303,13 @@ export const settings = {
 						onSplit={
 							insertBlocksAfter ?
 								( before, after, ...blocks ) => {
+									if ( ! canInsertSibling( 'core/list' ) ) {
+										// If it is not possible to insert a list
+										// don't split the block just try to insert pasted blocks after it.
+										// Unsupported blocks are ignored.
+										insertBlocksAfter( blocks );
+										return;
+									}
 									if ( ! blocks.length ) {
 										blocks.push( createBlock( 'core/paragraph' ) );
 									}
diff --git a/core-blocks/paragraph/index.js b/core-blocks/paragraph/index.js
index d7df95c47dc7ef..2fc2d217cc42aa 100644
--- a/core-blocks/paragraph/index.js
+++ b/core-blocks/paragraph/index.js
@@ -139,6 +139,7 @@ class ParagraphBlock extends Component {
 
 	render() {
 		const {
+			canInsertSibling,
 			attributes,
 			setAttributes,
 			insertBlocksAfter,
@@ -233,6 +234,13 @@ class ParagraphBlock extends Component {
 						} }
 						onSplit={ insertBlocksAfter ?
 							( before, after, ...blocks ) => {
+								if ( ! canInsertSibling( name ) ) {
+									// If it is not possible to insert block of the same type
+									// don't split the block just try to insert pasted blocks after it.
+									// Unsupported blocks are ignored.
+									insertBlocksAfter( blocks );
+									return;
+								}
 								if ( after ) {
 									blocks.push( createBlock( name, { content: after } ) );
 								}
diff --git a/editor/components/block-list/block.js b/editor/components/block-list/block.js
index 70eb8f82088c9e..0c8251c25d0b41 100644
--- a/editor/components/block-list/block.js
+++ b/editor/components/block-list/block.js
@@ -2,7 +2,7 @@
  * External dependencies
  */
 import classnames from 'classnames';
-import { get, reduce, size, castArray, first, last, noop } from 'lodash';
+import { get, reduce, size, castArray, filter, first, last, noop } from 'lodash';
 import tinymce from 'tinymce';
 
 /**
@@ -60,6 +60,7 @@ export class BlockListBlock extends Component {
 
 		this.setBlockListRef = this.setBlockListRef.bind( this );
 		this.bindBlockNode = this.bindBlockNode.bind( this );
+		this.canInsertSibling = this.canInsertSibling.bind( this );
 		this.setAttributes = this.setAttributes.bind( this );
 		this.maybeHover = this.maybeHover.bind( this );
 		this.hideHoverEffects = this.hideHoverEffects.bind( this );
@@ -123,6 +124,11 @@ export class BlockListBlock extends Component {
 		}
 	}
 
+	canInsertSibling( blockName ) {
+		const { canInsertBlockType, rootUID } = this.props;
+		return canInsertBlockType( blockName, rootUID );
+	}
+
 	setBlockListRef( node ) {
 		// Disable reason: The root return element uses a component to manage
 		// event nesting, but the parent block list layout needs the raw DOM
@@ -562,6 +568,7 @@ export class BlockListBlock extends Component {
 								id={ uid }
 								isSelectionEnabled={ this.props.isSelectionEnabled }
 								toggleSelection={ this.props.toggleSelection }
+								canInsertSibling={ this.canInsertSibling }
 							/>
 						) }
 						{ isValid && mode === 'html' && (
@@ -607,6 +614,7 @@ export class BlockListBlock extends Component {
 
 const applyWithSelect = withSelect( ( select, { uid, rootUID } ) => {
 	const {
+		canInsertBlockType,
 		isBlockSelected,
 		getPreviousBlockUid,
 		getNextBlockUid,
@@ -651,6 +659,7 @@ const applyWithSelect = withSelect( ( select, { uid, rootUID } ) => {
 		block,
 		isSelected,
 		hasFixedToolbar,
+		canInsertBlockType,
 	};
 } );
 
@@ -674,7 +683,8 @@ const applyWithDispatch = withDispatch( ( dispatch, ownProps ) => {
 			selectBlock( uid, initialPosition );
 		},
 		onInsertBlocks( blocks, index ) {
-			const { rootUID, layout } = ownProps;
+			const { canInsertBlockType, rootUID, layout } = ownProps;
+			blocks = filter( blocks, ( { name } ) => canInsertBlockType( name, rootUID ) );
 			blocks = blocks.map( ( block ) => cloneBlock( block, { layout } ) );
 			insertBlocks( blocks, index, rootUID );
 		},