SQL Server – Should Existing Files Be Overwritten for Additional Backups?

Not sure what you expect from partial backups, but they are only useful for read-only filegroups:

partial backups are designed for use under the simple recovery model to improve flexibility for backing up very large databases that contain one or more read-only filegroups.

Differential backups are a different story, but they are not used as you use them. They are supposed to speed to up a full restore by skipping over some log backups, not as a replacement for full backups.

Is rather unclear what you actually have. I guess you have a full backup and 130 differential backups. In such case you can restore the original full and then apply the last differential.

My suggestion is to implement a correct backup plan (full recovery mode, periodic full backup and frequent log backups). There are many examples how to do this. Start with reading Introduction to Backup and Restore Strategies in SQL Server. Go over some tutorials like the ones from SQL Server Backup Best Practices and Articles. Understand what you're doing, don't guess.

SQL Server allow for a database backup while the database is still being used. During a backup, most operations are possible;

for example

 INSERT
 UPDATE
 DELETE

statements are allowed during a backup operation.

Note : if you try to start a backup operation while a database file is being created or deleted, the backup operation waits until the create or delete operation is finished or the backup times out.

Importent Note :some Operations that cannot run during a database backup or transaction log backup include the following:

1) File-management operations such as the ALTER DATABASE statement with either the ADD FILE or REMOVE FILE options.

2)Shrink database or shrink file operations. This includes auto-shrink operations.

3)If you try to create or delete a database file while a backup operation is in progress, the create or delete operation fails.

3)If a backup operation overlaps with a file-management operation or shrink operation, a conflict occurs. Regardless of which of the conflicting operation began first, the second operation waits for the lock set by the first operation to time out. (The time-out period is controlled by a session time-out setting.) If the lock is released during the time-out period, the second operation continues. If the lock times out, the second operation fails.

Reference :msdn

You can refer this link it explain the scenario with nice examples