Hey Vysh,
On the whole, May to October makes for the only time one can reach Spiti without a lot of hassles. However, the months of July to September usually bring about torrential rains in the Spiti Valley so a lot of roads end up being blocked due to landslides. Make sure you read up well on the weather conditions if you're planning a trip in August. Hope you have a great trip! :)
These are the monsoon months when monsoon is in full swing in Kinnaur and it is better to avoid this time as the area is prone to landslides. However, Spiti valley itself remains fine at this time, so if you make plans do not forget to buy cartons of apple as they grow the best at this time.