Not by default, no.
Take a look at Apple's Supported Bluetooth Profiles for iOS, different devices support different profiles; however, general file transfer is not one of them.
Additional Information Here are some common uses for each of the
Bluetooth profiles listed above.
Hands-Free Profile (HFP 1.6) Hands-Free Profile allows Bluetooth
headsets and car hands-free kits to communicate with mobile phones.
Phone Book Access Profile (PBAP) Phone Book Access Profile allows the
exchange of Phone Book Objects between iPhone and other Bluetooth
It is often used between a car kit and a mobile phone to:
Allow the car kit to display the name of the incoming caller Allow the
car kit to sync the phone book so the user can initiate a call from
the car display Advanced Audio Distribution Profile (A2DP) Advanced
Audio Distribution Profile allows high quality audio (stereo or mono)
to be streamed from one device to another over a Bluetooth connection.
This is commonly used to play music wirelessly from iOS to a wireless
headset or a compatible car audio system.
Audio/Video Remote Control Profile (AVRCP 1.4) Audio/Video Remote
Control Profile allows remote control of playback functions on the iOS
device from compatible devices (commonly Bluetooth devices that use
the A2DP profile). The latest version of iOS supports the following
commands: pause, play, stop, next track, previous track, and browsing.
Personal Area Network Profile (PAN) Personal Area Network Profile
provides network connectivity over Bluetooth. With iOS this is common
with multiplayer games and Personal Hotspot on iPhone. See this
article for additional information.
iPod touch and iPad with iOS 4.3 or later support Internet tethering
from an iPhone with iOS 4.2.6 or later, through PAN.
Human Interface Device Profile (HID) Human Interface Device Profile
allows communication between iOS and Bluetooth keyboards.
Message Access Profile (MAP) Message Access Profile allows exchange of
messages between devices. It is mostly used for automotive hands-free
Note: iOS requires that the remote Bluetooth device support an
encrypted connection. When establishing a pairing record between your
iOS device and a Bluetooth device, you are also setting up encryption
between the two devices.