diff --git a/moto/awslambda/models.py b/moto/awslambda/models.py
index 316aa131b..318266abb 100644
--- a/moto/awslambda/models.py
+++ b/moto/awslambda/models.py
@@ -569,9 +569,9 @@ class LambdaFunction(CloudFormationModel, DockerModel):
 
             container = exit_code = None
             log_config = docker.types.LogConfig(type=docker.types.LogConfig.types.JSON)
+
             with _DockerDataVolumeContext(self) as data_vol:
                 try:
-                    self.docker_client.ping()  # Verify Docker is running
                     run_kwargs = (
                         dict(links={"motoserver": "motoserver"})
                         if settings.TEST_SERVER_MODE
@@ -608,23 +608,7 @@ class LambdaFunction(CloudFormationModel, DockerModel):
 
             output = output.decode("utf-8")
 
-            # Send output to "logs" backend
-            invoke_id = uuid.uuid4().hex
-            log_stream_name = "{date.year}/{date.month:02d}/{date.day:02d}/[{version}]{invoke_id}".format(
-                date=datetime.datetime.utcnow(),
-                version=self.version,
-                invoke_id=invoke_id,
-            )
-
-            self.logs_backend.create_log_stream(self.logs_group_name, log_stream_name)
-
-            log_events = [
-                {"timestamp": unix_time_millis(), "message": line}
-                for line in output.splitlines()
-            ]
-            self.logs_backend.put_log_events(
-                self.logs_group_name, log_stream_name, log_events, None
-            )
+            self.save_logs(output)
 
             # We only care about the response from the lambda
             # Which is the last line of the output, according to https://github.com/lambci/docker-lambda/issues/25
@@ -636,7 +620,24 @@ class LambdaFunction(CloudFormationModel, DockerModel):
             return resp, invocation_error, logs
         except docker.errors.DockerException as e:
             # Docker itself is probably not running - there will be no Lambda-logs to handle
-            return "error running docker: {}".format(e), True, ""
+            msg = "error running docker: {}".format(e)
+            self.save_logs(msg)
+            return msg, True, ""
+
+    def save_logs(self, output):
+        # Send output to "logs" backend
+        invoke_id = uuid.uuid4().hex
+        log_stream_name = "{date.year}/{date.month:02d}/{date.day:02d}/[{version}]{invoke_id}".format(
+            date=datetime.datetime.utcnow(), version=self.version, invoke_id=invoke_id,
+        )
+        self.logs_backend.create_log_stream(self.logs_group_name, log_stream_name)
+        log_events = [
+            {"timestamp": unix_time_millis(), "message": line}
+            for line in output.splitlines()
+        ]
+        self.logs_backend.put_log_events(
+            self.logs_group_name, log_stream_name, log_events, None
+        )
 
     def invoke(self, body, request_headers, response_headers):
 
diff --git a/tests/test_awslambda/test_lambda.py b/tests/test_awslambda/test_lambda.py
index f61ab9ea5..6be6e5e30 100644
--- a/tests/test_awslambda/test_lambda.py
+++ b/tests/test_awslambda/test_lambda.py
@@ -363,7 +363,8 @@ def test_invoke_function_from_sns():
     result = sns_conn.publish(TopicArn=topic_arn, Message=json.dumps({}))
 
     start = time.time()
-    while (time.time() - start) < 30:
+    events = []
+    while (time.time() - start) < 10:
         result = logs_conn.describe_log_streams(logGroupName="/aws/lambda/testFunction")
         log_streams = result.get("logStreams")
         if not log_streams:
@@ -375,13 +376,14 @@ def test_invoke_function_from_sns():
             logGroupName="/aws/lambda/testFunction",
             logStreamName=log_streams[0]["logStreamName"],
         )
-        for event in result.get("events"):
+        events = result.get("events")
+        for event in events:
             if event["message"] == "get_test_zip_file3 success":
                 return
 
         time.sleep(1)
 
-    assert False, "Test Failed"
+    assert False, "Expected message not found in logs:" + str(events)
 
 
 @mock_lambda
@@ -1306,7 +1308,7 @@ def test_invoke_function_from_sqs(key):
     assert msg_showed_up, (
         expected_msg
         + " was not found after sending an SQS message. All logs: "
-        + all_logs
+        + str(all_logs)
     )
 
 
@@ -1355,7 +1357,7 @@ def test_invoke_function_from_dynamodb_put():
     msg_showed_up, all_logs = wait_for_log_msg(expected_msg, log_group)
 
     assert msg_showed_up, (
-        expected_msg + " was not found after a DDB insert. All logs: " + all_logs
+        expected_msg + " was not found after a DDB insert. All logs: " + str(all_logs)
     )
 
 
@@ -1422,7 +1424,7 @@ def wait_for_log_msg(expected_msg, log_group):
     logs_conn = boto3.client("logs", region_name="us-east-1")
     received_messages = []
     start = time.time()
-    while (time.time() - start) < 10:
+    while (time.time() - start) < 30:
         result = logs_conn.describe_log_streams(logGroupName=log_group)
         log_streams = result.get("logStreams")
         if not log_streams: