First, by mathematical theory - yes (though in practice you can totally disregard this for modern hash functions and numbers achievable by computation in this universe). If the system doesn't store your password in plain-text or via reversible encryption, but instead hashes it (as is best practice) there are an infinity of potential passwords that will generate a hash that matches your hash (assuming an ideal cryptographic hash function).
For example, if your password was
password = '!78ghA,NJ58*#3&*' and they used a salted sha256 hash (note this should be key-strengthened with many rounds to make brute-force harder) with a 3-byte salt:
d7 35 e6 then your password hash would be
4fe5bbb74a21fb6d20785ce7fce1cd51d6fc87c5a55f65d75e8f37096ed54a53 (in python you can calculate this with):
>>> import hashlib
>>> salt = '\xd7\x35\xe6'
>>> password = '!78ghA,NJ58*#3&*'
>>> hashlib.sha256(salt + password).hexdigest()
Note this hash is 256 bits. So there are only 2256 ~ 1077 possible hashes. If the hash functions works well and selects hashes uniformly for all possible inputs (that is the hash function acts as a random oracle), once you try significantly more than 1077 hashes, some hashes will collide (provable by the pigeonhole principle). If you tried 1080 hashes, you should expect to find about 1000 passwords that match any given hash.
However, 1077 is an extremely large number; it's roughly the number of atoms in the observable universe. Or if every atom in the solar system (~1057) each calculated a hash every nanosecond it would take about 3000 years before you calculated 1077 hashes, so this isn't something you have to worry about in practice.
(If you tried 5 billion hashes per second for 100 years, and you had some very strong password the chance of randomly finding a different password that matches your 256-bit hash is 1 in 7.4 x 1057; this is like the odds of buying 7 powerball tickets on successive weeks and winning the jackpot the first 6 times and then playing a 7th time and matching 5 numbers (no powerball) to only win $1 million.)
Granted, this doesn't mean in practice your password is unique. I have personally run into a deployed Cisco VPN system in the wild that truncates a 12-character random password into 8 characters without notifying the user (I initially found out when I typed too fast and left off a character and it still worked; and then did trial and error). So as long as the first 8 characters are right, then you can type whatever you want after that and it will work.
This isn't isolated case either; e.g., Microsoft used to silently truncate Windows Live ID passwords to 16 characters though it now informs the user that it limits passwords to 16 characters:
Windows Live ID passwords were always limited to 16 characters—any additional password characters were ignored by the sign-in process. When we changed "Windows Live ID" to "Microsoft account," we also updated the sign-in page to let you know that only the first 16 characters of your password are necessary.”
It's also possible that a poorly designed authentication system will strip special characters out of the middle of your password. My friend found out that T-Mobile did this in 2011 when he used their poorly designed "forgot password" feature that sent him his password in plaintext which in his case stripped the special characters out of his password.