diff --git a/telethon/client/updates.py b/telethon/client/updates.py
index a789e77c..db5e3f3d 100644
--- a/telethon/client/updates.py
+++ b/telethon/client/updates.py
@@ -393,7 +393,11 @@ class UpdateMethods:
             # inserted because this is a rather expensive operation
             # (default's sqlite3 takes ~0.1s to commit changes). Do
             # it every minute instead. No-op if there's nothing new.
-            await self.session.save()
+            try:
+                await self.session.save()
+            except OSError as e:
+                # No big deal if this cannot be immediately saved
+                self._log[__name__].warning('Could not perform the periodic save of session data: %s: %s', type(e), e)
 
     async def _dispatch_update(self: 'TelegramClient', update):
         # TODO only used for AlbumHack, and MessageBox is not really designed for this
diff --git a/telethon/client/users.py b/telethon/client/users.py
index 615d97cf..4436cbbd 100644
--- a/telethon/client/users.py
+++ b/telethon/client/users.py
@@ -71,7 +71,11 @@ class UserMethods:
                             exceptions.append(e)
                             results.append(None)
                             continue
-                        await self.session.process_entities(result)
+                        try:
+                            await self.session.process_entities(result)
+                        except OSError:
+                            self._log[__name__].warning(
+                                'Failed to save possibly new entities to the session: %s: %s', type(e), e)
                         self._entity_cache.add(result)
                         exceptions.append(None)
                         results.append(result)
@@ -82,7 +86,14 @@ class UserMethods:
                         return results
                 else:
                     result = await future
-                    await self.session.process_entities(result)
+                    # This is called pretty often, and it's okay if it fails every now and then.
+                    # It only means certain entities won't be saved.
+                    try:
+                        await self.session.process_entities(result)
+                    except OSError:
+                        self._log[__name__].warning(
+                            'Failed to save possibly new entities to the session: %s: %s', type(e), e)
+
                     self._entity_cache.add(result)
                     return result
             except (errors.ServerError, errors.RpcCallFailError,