Can the major indice of an exporting nation decrease at the same time the currency devalues? In the UK, the FTSE100 is achieving record highs at the same time the currency (GBP) is depreciating rapidly. 

The media often correlate the two events by reporting that since the indice is made of 100 multinational companies who trade primarily in foreign currencies; a weak pound allows them to book larger profits after they trade back into the GBP domestically. 

However, can the FTSE100 decrease at the same time the currency also decreases in value or must the two levers always go in separate directions?
Yes, the FTSE 100 can decrease at the same time as the currency. It depends on what is causing the depreciation of the currency or what is affecting the FTSE at the time. It is hard to establish which way the effects are going in real time, however.

Suppose the currency is depreciating due to loose monetary policy. If the Marshall-Lerner condition is fulfilled, this means companies will export more and have more profit. In that case, we could expect the FTSE to go up. On the other hand, if the condition is not fulfilled, exports may actually go down and the FTSE with it. This is the case where the currency depreciation affects the FTSE.

On the other hand, the FTSE may affect the currency. If firms are facing a crisis or other problems, the FTSE goes down. This may prompt the central bank to attempt to depreciate the currency or it may prompt investments to flow out of the country, which depreciates the currency. The currency may depreciate for a while, before the monetary policy has any positive effects on the FTSE. The loss of investment will have a direct negative effect on the FTSE, as there is less capital supply and interest rates for firms may go up. Here again both the FTSE and the currency may go down. 

There could also be third factors that affect both the FTSE and the currency. Suppose the currency is depreciating because of other  factors, e.g. an increase in uncertainty. This may cause investments to flow out of the U.K. thereby depreciating the currency. Further, that same economic uncertainty and reduction of investment could hurt profits and therefore the FTSE could go down.

Moreover, a simultaneous crisis in the banking industry and a currency crisis is very possible. These are called "twin crises". See the 1999 Kaminsky and Reinhart paper in the AER for a reference. In that case, the FTSE and the currency can go down simultaneously. Twin crises are more relevant for fixed exchange rate regimes, however, so this possibility may not apply to the U.K. as much as to other countries.