Not unless you have a VASA provider for this storage configured. Otherwise you'd have to get this information directly from the storage.
For sure it's only possible if you can manage your storage from web client with VASA as mentioned eralier.
Also you can check vmkernel.log on host where you try to power on VMs. There should be some SCSI sence codes from storage array which should tell you that array report end of space on LUN.
As said above, you have to check either via VASA plugin or from SAN team, in case if the error occurred because of less free space in thin lun, you should see appropriate SCSI sense codes in vmkernel log of the server
Usually you can see in log something like "failed H:0x8 D:0x0 P:0x0 Possible sense data: 0xA 0xB 0xC"
In the above sense code
0xC is the plugin code which will give you information on the thin provisioned lun
following are the relevant code
06h Allocating more thin provision space. Device server is in the process of allocating more space in the backing pool for a thin provisioned LUN.
07h Thin provisioning soft-limit exceeded.
08h Backing pool for thin provisioned LUN is out of space.