While there is nothing limiting the computer from logically addressing all 264 addresses, the limitation happens due to the implementation of MBR-style partition headers. In the standard DOS-style MBR commonly implemented in most IBM-compatible computers, each 16-byte partition entry only has room for a 4-byte (32-bit) LBA sector address.
Because this only allows one to address at most 232 sectors, and given each sector is 512 bytes, the maximum possible range that can be accessed for any MBR-style partition is 232 x 512 bytes, which equals 2048 GiB, or 2 TiB. Using a drive with a larger sector size (e.g. 4096-bytes) will extend the maximum range to 16 TiB, but this requires a computer and operating system compatible with larger sector sizes other than the standard 512-byte sector.
This is also briefly discussed in the Wikipedia article for logical block addressing:
The current 48-bit LBA scheme, introduced in 2003 with the ATA-6 standard, allows addressing up to 128 PiB. [...] However, the common DOS style Master Boot Record (MBR) partition table only supports disk partitions up to 2 TiB in size. For larger partitions this needs to be replaced by another scheme, for instance the GUID Partition Table (GPT) which has the same 64-bit limit as the current INT 13h Extensions.
Indeed, in GPT-style partitions, there are two 8-byte fields specifying the starting and ending logical block address for that partition, extending the addressable range to 264 sectors. Similar calculations can be performed as above to compute the maximum partition size for a disk by multiplying the sector size (usually 512 or 4096 bytes) by 264.
As a final note, GPT partition tables also differ slightly in their implementation/layout. MBR-style partitions store both the beginning and ending sectors in CHS (3-bytes) format, as well as the beginning sector and size of the partition in LBA format (4-bytes each). GPT does away with this convention, and instead stores two LBA (8-byte) values, representing the beginning and ending sectors, inclusive, for that partition.